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Geometric frustration: the case of triangularantiferromagnets

Pierre-Éric Melchy

To cite this version:Pierre-Éric Melchy. Geometric frustration: the case of triangular antiferromagnets. Data Analysis,Statistics and Probability [physics.data-an]. Université de Grenoble, 2010. English. tel-00537747

THÈSE thèsePour obtenir le titre de

de docteur de l’université de GrenobleSpécialité : physique (matière condensée)

Arrêté ministériel : 7 août 2006

Présentée et soutenue publiquement par

Melchy Pierre-Éricle 8 octobre 2010

Frustration géométrique:le cas des antiferromagnétiques

triangulaires

Thèse dirigée par Mineev Vladimir et codirigée par Zhitomirsky Mike

jury

M. Holdsworth Peter École Normale Supérieure de Lyon RapporteurM. Honecker Andreas Université de Göttingen RapporteurMme Lacroix Claudine Institut Néel, CNRS PrésidenteM. Zhitomirsky Mike INAC, CEA

i

Abstract

This doctoral dissertation presents a thorough determination of the phase diagrams of

classical Heisenberg triangular antiferromagnet (HTAF) and its anisotropic variants based on

theoretical and numerical analysis (Monte Carlo). At finite-field HTAF exhibits a non-trivial

interplay of discrete Z3 symmetry and continuous S1 symmetry. They are successively broken

(discrete then continuous) with distinct features at low and high fields: in the latter case the

ordering is along transverse direction; in the former case an intermediate collinear phase is

stabilised before 120-degree structure is. Due to zero-field behaviour, transition lines close at

(T, h) = (0, 0).

Single-ion anisotropy is here considered. Easy-axis HTAF for moderate anisotropy strength

0 < d 6 1.5 possesses Z6⊗S1 symmetry at zero-field which induces triple BKT-like transitions.

At finite field the symmetry is the same as for HTAF: both thus share the same symmetry-

breaking pattern. Yet specificities can be observed in the easy-axis system: splitting of

zero-temperature transition at one-third magnetisation plateau, reduction of the saturation

field.

Easy-plane HTAF belongs to the class of universality of XY triangular antiferromagnet:

it thus interesting to start with this system. Zero-field behaviour results from the breaking

of Z2 ⊗ S1 symmetry, where the discrete component is an emerging chiral symmetry. An

intermediate magnetically chiral ordered phase exists which extends to finite-field where the

symmetry is Z2⊗Z3. The upper limit of this intermediate phase along field axis is a multicrit-

ical point at which transition lines are inverted. Above, the intermediate phase is a collinear

phase. At high field the compound symmetry is broken as a whole Z6.

ii

Résumé

Cette thèse de doctorat présente la détermination théorique et numérique (Monte Carlo)

du diagramme de phase du système classique antiferromagnétique de Heisenberg sur réseau

triangulaire (HAFT) et de ses variantes anisotropes. Sous champ HAFT présente une intri-

cation non triviale des symétries discrète Z3 et continue S1. Elles sont successivement brisées

(discrète puis continue) selon des modalités différentes à champ fort et modéré : dans ce cas-là

l’ordre a lieu selon la direction transverse ; dans ce cas-ci une phase colinéaire intermédiaire

est stabilisée avant la phase à 120 degrés. Du fait du comportement à champ nul les lignes de

transitions se terminent à (T, h) = (0, 0).

L’anisotropie mono-ionique est ici considérée. HAFT avec anisotropie d’axe facile pour

une anisotropie modérée, 0 < d 6 1.5, possède une symétrie Z6 ⊗ S1 à champ nul, qui induit

une triple transition BKT. Sous champ, la symétrie est identique à HAFT : les deux partagent

donc le même scénario de brisure de symétries. Le système anisotrope présente toutefois des

spécificités ; séparation de la transition à température nulle au champ de tiers d’aimantation,

réduction du champ de saturation.

HAFT avec anisotropie de plan facile appartient à la classe d’universalité de XY AFT il

est donc intéressant de commencer par ce système-ci. Le comportement à champ nul résulte de

la symétrie Z2⊗S1 où la composante discrète est une symétrie chirale émergente. Une phase

intermédiaire chirale magnétiquement désordonnée est stabilisée ; elle se prolonge sous champ,

où la symétrie est réduite à Z2 ⊗ Z3, jusqu’à un point multicritique auquel les transitions

s’inversent. Au-dessus de celui-ci la phase intermédiaire est colinéaire. Sous champ fort la

symétrie composite se brise comme une symétrie Z6 unique.

Acknowledgement

Preparing a doctorate couldn’t have been possible without the presence of manypersons throughout the years as it is mere a certain achievement of a part of my lifethan a three-year challenging experience.

First I should thank my supervisor M. Zhitomirsky whose ruthless scientific demandprevented me to lean towards any kind of lazy thinking or lack of rigour. This is a rarequality in an inflating research world, quality for which I am really grateful to him. I’dalso like to thank other senior researchers who were around during these years at CEA(thanked for its material support): J. Villain and V. Mineev for their mentoring andyet distant presence; X. Waintal, and M. Houzet for the fresh blood they are injectinginto the theory group. As said there are many more persons who helped me alongmy way. Professors and tutors ignited and fueled my eager curiosity for understandingthat has kept on growing with knowledge alongside an ever more acute perception of itsstunning limitations requiring a tireless conquest of its frontier. In physics I’d like tothank F. Mila, B. Kumar, M. Kläui, P. Martin, A. Georges, M. Mézard, A. Aspect, J.Dalibard, J.-L. Basdevant, and S. Laurens. In mathematics I can’t help being gratefulto J. Lannes, J. Chevallet, N. Tosel, and C. Viano. Other names arise in my mindfrom which I’d like to extract only two for this page not to turn into an endless tribute:J.-P. Dupuy for his enlightening conversations, for his passionate and yet rational tohis fingertips commitment to the criticism of an erring science; S. Robert for his brightand reinvigorating approach to epistemology.

Beside these persons I cannot but warmly thank my colleagues turned friends andfirst of all Raphaël who has shared much of the hard time and quite a few joyful momentsof these last three years; Sean, who left a year ago, helped me a lot in many regardseven though I may wish we had had more time to share ; Martin brought a cheerful andenthusiastic youth. Of course I met other so-called young researchers (PhD studentsand postdocs) whose encounter was appreciable: may they be thanked even withouttheir naming. Yet I’d like to address personal thanks to a youngster whose passion forphysics and joyful character brought quite an appreciable refreshment: Vincent.

For all their precious and enriching friendship throughout the years without whichit would undoubtedly have been much harder if not impossible altogether I rejoiceaddressing with all my heart my very thanks to Claire, Delphine, Michel, Philippe,

iii

iv

Arnaud, Philippe, Jean-Paul, Arnaud, and François-Xavier. Their joyful warmth, theirsupportive presence, their challenging pieces of advice, their simple sharing the ups anddowns are parts of what has helped me up. Persons I am profoundly grateful to andlovingly thankful to for their countless efforts to foster my development, for their beinginfinitely open-minded regarding my choices, wills and desires, for their never lackingsupport whatever the dire situation they, I, or we may be facing are my parents andmy brother.

Last I’d like to thank these inspiring figures that enlighten my sky, all these greatpersons who make me proud of being called a human-being, whether they may bescientists, philosophers, artists or other great achievers, famous or not, of what can becalled the best part of our humanity. Out of my personal sky I could name quite a fewof these stars that lead my course. However I limit myself to the one who stood by myside throughout the writing of this dissertation and represents a model for any writerfor his avoiding any superfluous note: Johann Sebastian Bach.

Contents

1 Introduction 11.1 Symmetry breaking in 2d . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Triangular antiferromagnet: historical perspective . . . . . . . . . . . . 9

2 Heisenberg Triangular Antiferromagnet (HTAF) 132.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Brief review of zero-field behaviour . . . . . . . . . . . . . . . . . . . . 152.3 Finite-field behaviour: symmetry discussion . . . . . . . . . . . . . . . 182.4 Finite-field behaviour: numerical determination . . . . . . . . . . . . . 23

2.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.4.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 HTAF with easy-axis single-ion anisotropy 313.1 Symmetries at zero field . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Real-space mean-field approach . . . . . . . . . . . . . . . . . . . . . . 343.3 Zero-field behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Finite-field behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 HTAF with easy-plane single-ion anisotropy 454.1 XY triangular antiferromagnet at zero-field . . . . . . . . . . . . . . . 464.2 Low fields: competing Z2 and Z3 symmetry breaking . . . . . . . . . . 494.3 Breaking of Z6 symmetry at high fields . . . . . . . . . . . . . . . . . . 534.4 Easy-plane HTAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Conclusion 59

A Single-ion anisotropy 63

B Real-space mean-field theory 65B.1 Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.1.1 Single spin in an external field . . . . . . . . . . . . . . . . . . . 65

v

vi CONTENTS

B.1.2 Lattice model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66B.2 Anisotropic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

B.2.1 Classical spins problem . . . . . . . . . . . . . . . . . . . . . . . 67B.2.2 Quantum case S=1 . . . . . . . . . . . . . . . . . . . . . . . . . 68

C Zero-field upper transition in mean-field treatment: sign of γ2 71

D Finite-size scaling 75D.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

D.1.1 Geometry and boundary conditions . . . . . . . . . . . . . . . . 75D.1.2 Alteration of singularities in finite systems . . . . . . . . . . . . 77D.1.3 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

D.2 Expressions of direct interest in this study . . . . . . . . . . . . . . . . 78

E Monte Carlo algorithms 81E.1 Back to basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81E.2 Out of traps: over-relaxation . . . . . . . . . . . . . . . . . . . . . . . . 84E.3 Estimating errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

F French summary 89

Chapter 1

Introduction

Condensed matter physics deals with matter in its condensed form: what doesthis tautology state? In this context condensed can be defined as held together thanksto internal interactions. It means that this branch of physics doesn’t investigate el-ementary entities but the collective behaviour of interacting particles. Its name wascoined rather recently (1967) by P. W. Anderson and V. Heine when they renamedtheir Solid-state Theory laboratory at Cambridge, UK, Theory of Condensed Matter.From a perspective in terms of macroscopic physical properties, they made the focusevolve towards the underpinning phenomenon, the grounding effect of which extendsbeyond the sole solid state, namely many-body interactions and the induced collectivephenomena measurable either at a microscopic or macroscopic level. One of the moststriking feature of collective phenomena is phase transition. Phase transitions are acommonly experienced fact — anybody cycling in winter does know that water in itssolid form, also known as glaze when covering the ground, has quite distinct properties.Yet their understanding and their description is far less straightforward and has beenfostering the development of theories and experiments by physicists for generations.They arise whenever there are competitive processes governing the equilibrium of asystem: varying external parameters such as temperature, pressure, magnetic field, itis then possible to change the equilibrium configuration. The trouble in condensedmatter physics stems from the hardship to analytically describe systems with morethan two interacting particles: yet any realistic system such as this very sheet of paperconsists of several billions of billions of atoms — this is no reason to conclude that thisthesis is intractable: another important characteristic of condensed matter physics is tolook at systems at the relevant scale and I doubt the atomistic one is the right choicefor this piece of work! To overcome this hardship it has been necessary to develop waysof treating systems with a huge number of particles in a tractable way: this is whatstatistical mechanics can do. It makes use of the observation that physical properties ofa given system can be correctly described by probabilistic distributions. Boltzmann can

1

2 CHAPTER 1. INTRODUCTION

without hesitation be named the father of this revolutionary approach.1 Thanks to thisrevolutionary viewpoint it has been possible to describe macroscopic properties of mat-ter from microscopic underpinning phenomena. The formalism of statistical mechanics,however originally devised in a classical language, could fully be extended to quantumcontext with a straightforward correspondence. With this toolbox in hand physicistscould further develop the understanding of collective phenomena, among which criticalphenomena that occur at phase transitions.

In a way or another various modern descriptions of critical phenomena are builtupon the observation criticality is characterised by a loss of scale hierarchy: about phasetransition the system look the same at different scales — this auto-similarity is coined asfractal. In other words details don’t matter. An early understanding of this statementtook the form of mean field as enunciated by Weiss in 1907 [128]: this approach consistsin considering for each particle the interactions with its neighbours and the environmentat the level of mean values. However simplistic such a treatment may seem it hasrevealed not only fruitful to orientate intuitions but also exact in certain limits (roughlyspeaking in high dimensions for short-ranged interactions). Another development of thisidea was proposed by Landau with his phenomenological description of second-orderphase transitions2 based on a symmetry analysis of the system. Landau’s discussion isin fact two-fold. On the one hand he explicited symmetry rules governing a second-orderphase transition: at a continuous transition there must be a group-subgroup relationbetween the symmetry groups of each phase on both sides of the transition. This leadsto a classification of possible continuous phase transitions given a model with a specificsymmetry group. On the other hand he proposed an hydrodynamic-like developmentof the free energy functional in terms of successive powers of the order parameter andits gradient (which is possible at a continuous transition as the order parameter goesto zero3). Both sides of this reasoning work together as the development of the freeenergy functional introduces terms that must respect the symmetry of the model. Thisfruitful theory elegantly circumvents a major difficulty of statistical mechanics, namelythe explicit calculation of partition function given a specific Hamiltonian. A laterdevelopment fully pushing this idea of scale-invariance and discarding irrelevant detailswas the development during the 1960’s of renormalisation group by Kadanoff [41] andWilson [129, 130]. The underlying idea is to describe the model at larger and larger

1Boltzmann’s epitaph reads: S = k logW .2After Ehrenfest’s first classification of phase transitions according to continuity properties of the

derivatives of free energy (in that classification an n-th order transition is a transition at which firstdiscontinuity occurs for the n-th derivative), modern classification distinguishes first-order transitionscharacterised by the existence of non-zero latent heat from second-order ones that are continuous(without any latent heat) and associated with a diverging correlation length at the transition. Infinite-order transitions exist as well such as Berezinskii-Kosterlitz-Thouless transition that is dealt with inthis dissertation.

3Extensions of Landau’s formalism to certain first-order transitions can be done as presented forexample in [119].

3

scales using a coarse-graining approach: during this transformation, that is called aflow, coupling constants of the model undergo changes that constitute a semi-group(hence the name that mathematically speaking is not exact) with certain constantsflowing to zero, which means they don’t play any role for the criticality of the system.With this viewpoint it was then possible to introduce classes of universality: such aclass groups various systems sharing common relevant interactions and consequentlycommon symmetries. Each class can then be defined by a set of critical exponents thatdescribe how quantities of interest such as specific heat behave in the vicinity of thetransition. Most of physicists’ efforts in the analysis of phase transitions has thus beenthe determination of the class of universality which the model they investigate belongsto and in another direction the attempt to describe all possible classes of universality.

The latter effort has undergone a dramatic change with conformal field theory. Thistheory is based on the fact that critical theories are not only invariant under changesof scale as previously introduced with the renormalisation group but also under theaction of conformal transformations4. If conformal invariance doesn’t bring anythingnew in dimensions d > 3, for d = 2 it does bring new constraints that should enable tocatalogue bidimensional critical phenomena. This statement singles out the specificityof dimension 2. 2 d is in many regards specific as it offers a wealth of models with exoticcritical behaviours that cannot be found in other dimensions; some of them even evadetreatment with methods such as mean-field approximation, renormalisation group, andother methods perfectly working at higher dimensions. A way to grasp this specificityof bidimensional models is to think of their topological properties: as can readily beobserved continuous deformations that are the pictorial way to glimpse at topologyare far more restricted in 2d than in higher dimensions where it is often possible tocircumvent a singularity using the extra dimensions. On the other side most methodsused in 1d systems are specific to this dimension with the extreme constraint of thedimensionality that induces specific collective excitations.

Beside these theoretical achievements, numerical physics has grown in importanceand played quite crucial a role in the understanding of critical phenomena. As pre-viously shortly alluded to one major hindrance in statistical mechanics is the actualhandling of partition function. Numerical methods can precisely help overcoming iteven without any explicit calculation of the partition function at stake — in the endthe interesting elements are physical observables rather than mathematical devices usedto built up theories. One major player in the field of numerical statistical physics isMonte Carlo procedure in its various variants. Undoubtedly the algorithm proposed byMetropolis and coworkers [78] fostered the emergence of the field, which has been fur-ther boosted by the exponential growth of computing facilities pushing further away thebalking limitations of numerical simulations. In a word the idea behind these methods

4The group of conformal transformation is the subgroup of coordinate transformations that leavethe metric invariant up to a scale factor, ie that preserve the angle between two vectors.

4 CHAPTER 1. INTRODUCTION

is to astutely explore the phase space to catch a faithful glimpse of the system underscrutiny: this is achieved through a Markov chain5. Numerics has thus grown as thethird pillar on which modern statistical physics stands.

With this well equipped toolbox in hand it becomes possible to deal with phys-ical models among which magnetic systems constitute one of the most appreciatedplayground thanks to the variety of models that can be both theoretically devisedand experimentally studied. An incomparable advantage of magnetic systems is in-deed that many experimental techniques are available to both probe macroscopic andmicroscopic properties: from bulk thermodynamical measurements (specific heat, sus-ceptibility, etc.) to local probing (atomic force microscopy, muon spin resonance), withsuch fine structural investigation tools as neutron scattering experiments (elastic andinelastic scattering, polarised or unpolarised neutrons, spin echo, etc.). Furthermore itis most of the time possible to write models accurately describing these spin systemsor in the reverse way certain models initially theoretically devised and studied haveproven relevant for the description of real compounds. For sure magnetic materials areless clean and less tunable than artificial magnetic crystals obtained in quantum optics.The latter are however still out of the energy range of interest and therefore remainpromising experimental toys not yet at their full maturity [14].

As said condensed matter deals with collective phenomena and consequently coop-erative behaviours. In quite a few circumstances cooperation can lead to an extremecase which is frustration. Roughly speaking (more precise a definition is proposed inthe following) frustration occurs whenever it is impossible for all interacting entities tosimultaneously reach an optimum. In spin systems the concept was formally introducedin the 1970’s. With the rough picture above proposed it can readily be understood thatthis concept is relevant to a huge variety of problems in statistical mechanics, and evenfar beyond, as such fields as econophysics or sociophysics flourish. As previously pointedout magnetic systems provide an actual playground to deal with abstract concepts andfrustration is no exception: various compounds indeed embody frustration and makeit possible to confront models with experiments and reversely to seek inspiration in re-ality. Frustrated magnets offer quite an interesting playground to test various excitingconcepts beside the release (or not) of frustration such as spin liquids, exotic phasetransitions, quantum criticality, etc.

With this background in mind it is now possible to proceed to a more precise intro-duction of grounding concepts and questions supporting this dissertation. Symmetriesin 2d is the topic of Sec. 1.1. Then a thorough introduction to frustration is proposedin Sec. 1.2. Last an historical perspective on the family that models studied in the fol-lowing chapters of this dissertation belongs to, namely antiferromagnetic spin sytemson a triangular lattice, is proposed in Sec. 1.3.

5A Markov chain is a random process such that next step only depends on the current state: it ismemoryless, which makes it perfectly suited for numerical simulations.

1.1. SYMMETRY BREAKING IN 2D 5

1.1 Symmetry breaking in 2d

When dealing with phase transitions in general the question of symmetry breakingnaturally comes out. In 2d this issue acquires a dramatic specificity. In 1966 Merminand Wagner demonstrated that the breaking of a continuous symmetry in bidimen-sional systems is impossible at any finite temperature. As a consequence second-orderphase transition in spin systems with continuous rotational symmetry, as is the caseof isotropic models of spins with more than one component in zero field, is excluded.This statement doesn’t end the story. Berezinskii on the one hand [10, 11], Kosterlitzand Thouless on the other hand [61] argued that despite its continuous symmetry XYmodel on the square lattice does undergo a finite-temperature transition. No symmetry-breaking is associated with this transition but a dramatic change in the behaviour ofstable topological defects. Such a transition is an example of an infinite-order transi-tion; this one is referred to as a BKT transition. The discussion of phase transition interms of topological defects was formalised by different physicists [82, 76, 75, 79] in thelate 1970’s. It is based on homotopy groups: these groups concentrate the sufficientinformation to describe topological properties of objects. The simplest one, the fun-damental group, π1, describes how closed loops in a topological object evolve under acontinuous deformation. For example any closed loop on the sphere S2 can be shrunkto a point, hence the fundamental group of the sphere is the trivial group: π1(S2) = 0.On a circle the situation is less simplistic: some closed loops can twine around the cir-cle without being shrinkable to a single point; moreover these can twine several time,which means an integer can be associated to loops which is its winding number (zero incase of a shrinkable loop). This makes it understandable that the fundamental groupof a circle is the group of integers: π1(S1) = Z. This theory enables the handlingof topological defects which exist in physical systems. These topological defects caninduce phase transitions, which means it is possible to describe phase transition cal-culating the homotopy group of the symmetry group of the Hamiltonian. XY modelexemplifies this approach: the symmetry group of the Hamiltonian is S1; as seen aboveπ1(S1) = Z. As a consequence XY system admits stable point defects that are integervortices. The observation put forward by Berezinskii, Kosterlitz, and Thouless is thatvortices are bound into pairs of vortex-antivortex at low temperature and are free athigh temperature, which means a transition occurs in between: this binding-unbindingtransition is BKT transition. As a consequence of above mentioned Mermin-Wagnertheorem the order settling below the transition is not a long-range order but rather aquasi-long-range order. This low-temperature phase is hence a soft massless phase withpower-law decaying correlation functions and continuously varying critical exponentsin contrast to what happens in Ising model with its massive low-temperature phase.

An interesting question is then the evolution from one model to the other. A wayto study it consists in investigating discrete Abelian models with Zp symmetry. Besidethis theoretical motivation, the study of such models is motivated by the melting of

6 CHAPTER 1. INTRODUCTION

bidimensional crystals that are governed by discrete rotational symmetry (p = 2, 3, 4, 6are of experimental relevance). If an XY model perturbed by Zp terms was studied byJosé and coworkers in their seminal 1977’s work, first specific studies of pure Zp systemscame slightly later [28, 17, 27]. These works showed that there is a critical value pcsuch that for p 6 pc two massive phases exist as in Ising model whereas for p > pcbetween these two massive phases an intermediate critical massless phase (or quasiliq-uid) emerges the lower limit of which tends to zero as p goes to infinity in agreementwith Mermin-Wagner prescription. An even more striking result has been obtainedon p-state clock model, aka Zp models: the existence of an extended universality [65].Above a certain temperature Teu for p > 4, thermodynamical properties are provedidentical to those of the continuous model p =∞. In particular for p > 8 this collapsestarts in the intermediate quasiliquid phase, which implies that the upper transition isa real BKT transition. It constitutes an example of an emergent symmetry.

Another example of an emergent symmetry in bidimensional systems is the one of anextra discrete degeneracy in certain bidimensional systems with a continuous symmetry[122]. This phenomenon revealed by Villain arises as a consequence of multi-q structure.Let’s introduce a spin structure Si = u cos q·ri+v sin q·ri where u and v are orthogonalunit vectors and q is an ordering vector6. In case this structure describes all groundstates, which is the case with spins of dimension n = 2 or 3 (unless the ordering vectorq lies at special positions within Brillouin zone in this latter case), this formula showsthat an extra discrete degeneracy exists for spins of dimensionality n = 2 as soon as thestar7 of q consists in more than one vector and for n > 3 if the star is not reduced toq,−q. Let’s explicit this assertion in the former case: changing q into −q changessine into its opposite.As the vectors are bidimensional there is no direct continuoustransformation to change q structure into −q structure. The existence of this extradegeneracy makes it possible for the system to order at finite temperature despite itsoriginal continuous symmetry. A widely discussed example is the case of XY model onthe triangular lattice with its emergent chiral order that is presented in Chap. 4.

1.2 Frustration

Frustration is a concept that was formally introduced in the context of spin glasses[121, 123] even though earlier studies dealing with frustrated magnets exist [125, 114,44, 2]. A characteristic property of frustrated systems, namely their extensive entropyat zero temperature, was discussed as early as 1935 by Pauling [96] in water ice. Frus-tration can be defined as the impossibility to minimise all individual interaction terms

6q is obtained as a vector minimising with respect to k the Fourier transform∑

j Jij cos k · (ri−rj)where (Jij) is the set of bilinear exchange constants defining the Hamiltonian H =

∑

〈ij〉 Si · Sj .7The star of a vector k is the set of inequivalent vectors generated by the action of the lattice

symmetry group on the vector k.

1.2. FRUSTRATION 7

at the same time, may it be due to randomness, to geometric constraints, or to com-peting interactions. After the formal introduction of the term frustration, frustratedmagnets without randomness were studied for a while for their connection with spinglass. Yet it soon became obvious that these magnetic systems that had been stud-ied for some of them before the hype about spin glass could bring more. Thanks tothe diversity of experimental methods available to study magnetic systems, this fieldof research has experienced a continual and vivid cross-fertilisation between theoreti-cians and experimentalists. Investigations on frustrated magnetic materials have hadimplications far beyond magnetism itself. Indeed their highly degenerate ground statemanifold, their possibly non-collinear or incommensurate order, the assumptive spinliquid, and their possibly novel phase transitions offer a playground to investigate chal-lenging and exciting fundamental questions both in classical and quantum systems.Regarding quantum systems such questions as the link between cuprates supraconduc-tors and 2d quantum frustrated antiferromagnets [3] or as deconfined quantum criticalpoint, which is a new paradigm to describe phase transitions beyond Landau-Ginzburg-Wilson paradigm encompassing transitions between phases with no symmetry relation[104, 105], have renewed the vivacity of research on this topic. Interestingly variousanalytical, numerical and experimental techniques have been used to study frustratedmagnetic systems: analytical developments à la Onsager, Landau-Ginzburg treatmentand more generally analysis of symmetry and topological properties, mean-field tech-niques, renormalisation group apparatus, high- and low-T series expansions, MonteCarlo simulations, experimental investigations. In certain cases some of these differ-ent approaches may be at odds such as exemplified by the opposition between certainrenormalisation group methods (ε = d−2 development of a non-linear σ model) on theone hand and Monte Carlo simulations and topological discussions on the other handto describe classical frustrated Heisenberg spin systems [4, 49]. Renormalisation groupapproaches, Monte Carlo simulations and experimental measurement do not yield aconsistent picture of such systems, which is the illustration how non-trivial the criticalbehaviour of frustrated magnets is. It also points out the necessity to carefully under-stand the limitations of the techniques that are used in order to identify the origin ofsuch mismatches; hence a better insight into these techniques can be gained. Frustratedmagnetism has been at the heart of much highlighted research of the past thirty yearsas is the case with cuprates high-Tc supraconductors, Josephson junction arrays, multi-ferroics, etc. The most recent topic creating a real hype in this field was the descriptionof pseudo magnetic monopoles in spin ice systems [18]. Frustration can be studied ininsulating crystals as well as in metals or in disordered systems. Hereafter we consideronly the case of insulating crystals.

In insulating crystals the relevant picture to understand magnetism is the one ofisolated spins located at vertices of a lattice. From the Hubbard model one can derivelocalised-spin interaction Hamiltonians: depending on spin dimensionality, they areIsing (1d spin space), XY (2d) or Heisenberg (3d) models. The simplest cases consist

8 CHAPTER 1. INTRODUCTION

in bilinear interaction terms. In such cases the competition of interactions inducingmagnetic frustration can stem either from a competition between different interactionpaths (typically between nearest-neighbours and next-nearest neighbours) or from thetopology of the lattice. Widely studied examples of the former case are J1 − J2 modelon the square lattice, spin ladders, among others [21, 81]. In the latter case the frus-tration is said to be geometric. Geometrically frustrated magnets build up a major anddiverse group of magnets. Geometrically frustrated systems can typically be built withtriangular elementary plaquettes that can arrange either on a corner sharing pattern(kagome lattice) or on an edge sharing one (triangular lattice), in 2d or 3d as well.Another common building block is tetrahedron (corner sharing tetrahedra can form theso-called pyrochlore lattice). Common frustrated lattices comprise the 2d triangularlattice, kagome lattice, fcc, and pyrochlore among other ones. Numerous materials inthis class exist [36]: anhydrous alum, jarosites, pyrochlores, spinels, magnetoplumbites,garnets,etc. Geometrically frustrated spin systems enable us to study frustration invery simply formulated models and to deal with non-trivial topology questions. Indeedan important characteristic of these systems is the nature of the order parameter whichcan be such an object as a matrix of SO(3), or a complex vector with S1 ⊗ Z3 sym-metry group. Homotopy theory then yields non-trivial topological excitations, whichmay lead to exotic phase transitions. The identification of a new class of universalityis however a tricky issue due to the non-trivial critical behaviour of frustrated spin sys-tems and the complicated order parameter symmetry group. A famous example of sucha difficulty is provided by the twenty-year-long controversy about the nature of thephase transition in Heisenberg antiferromagnet in stacked triangular crystals. Earlyclaims of a new universality class based on two-loop renormalisation group analysisand Monte Carlo simulations appeared [47, 48]. Various simulations and theoreticalanalysis were then published. Using a non-linear σ model Azaria and coworkers [5]claimed the transition should pertain to O(4) universality class if it were not first-orderor mean-field tricritical. Tissier and coworkers published an extensive non-perturbativerenormalisation group study of frustrated spin systems in dimension between two andthree: a consequence of their study is that the transition of Heisenberg stacked trian-gular antiferromagnet is first-order; they further argue that the reason why numericalsimulations stalled around the transition and identified it as second-order with newcritical exponent is the existence of a region in the flow diagram where the flow is slow,inducing a very weak first-order character [116, 118]. Last Ngo and Diep published acareful Monte Carlo simulation using both advanced techniques and very large clustersto support the first-order character of the transition [88]. Similarly in this work wepresent results that firmly stand against a new universality class for the breaking ofS1 ⊗ Z2, which agree with the detailed analysis presented on fully frustrated XY spinsystems [37].

An important characteristic of frustrated magnets is the extensive degeneracy oflow-energy modes, which induces extensive entropy at zero temperature. Such a highly

1.3. TRIANGULAR ANTIFERROMAGNET: HISTORICAL PERSPECTIVE 9

degenerate ground state manifold can induce some long sought states such as spinliquids or spin glasses without any randomness, which is another reason why so mucheffort has flowed into research on frustrated spin systems. Spin liquids can be definedas gapped spin systems with a finite correlation length at zero temperature [80]. Onegood example of this is kagome antiferromagnet; yet its experimental realisation is stilllacking: the grail of a perfect spin-1/2 kagome system still seems far away. A dramaticconsequence of such a degeneracy is the possible appearance of extra soft modes, at leastat T = 0 as is the case for XY classical spins on the triangular lattice or for Heisenbergspins on the triangular lattice. This is however a fragile feature that is quite sensitiveto various perturbations and makes it even harder to observe, all the harder as anorder by disorder phenomenon [124], for example induced by thermal fluctuations, canoccur. It is by no way a systematic phenomenon in geometrically frustrated systems asHeisenberg pyrochlore and four-component spin system on kagome lattice show: bothremain disordered at low temperature [86].

Yet ways to remove this accidental continuous degeneracy exist. First, thermalfluctuations are expected to induce an order by disorder phenomenon as pointed out byVillain and collaborators at the very beginning of 1980’s [124]; however such an orderingmay occur in certain cases only at higher order than the second one as discussed bySheng and Henley [106]. Quantum fluctuations are another way to reduce degeneracyand induce order that will not be developped in this work. Last anisotropy changessymmetry, which may induce ordering: this point is the object of a large part of thework here presented.

1.3 Triangular antiferromagnet: historical perspec-tive

When considering geometric frustration the simpler system to come to the mind isan antiferromagnetic model on the triangular lattice. With their stunning simplicitytriangular antiferromagnets have been occupying physicists for several decades. As thisdissertation deals with classical systems the historical perspective here proposed leanstowards classical systems even though their quantum counterparts do present severalinteresting features. Reviewing what has been done on triangular antiferromagnetsclearly shows various research on these systems have gone different paths. After thehistorical exact solution of Ising models [125, 126] completed by the demonstration thismodel belongs to Ising universality class with a Z6 symmetry-breaking term [1] with anupper transition at T = 0 and further refinements in the discussion of Z6 symmetry-breaking as reinforced by the introduction of next-nearest-neighbour couplings [63, 33],much effort has been devoted either to quantum Ising models with and without field (inthe latter case, a transverse field enables to have a glimpse at other interesting models:the dual model is a Z2 gauge model that is equivalent to quantum kagome antiferro-

10 CHAPTER 1. INTRODUCTION

magnet, system expected to exhibit a spin-liquid phase) or to stacked triangular Isingantiferromagnet. The investigation of stacked triangular antiferromagnets in case ofXY and Heisenberg models has also gathered much attention, probably thanks to ex-perimental realisations of these models [23, 49]. Most investigated layered compoundsdiverge from 2d models as they are in fact chains weakly coupled in a triangular lattice,and thus exhibit 3d ordering properties of quasi-1d objects. A large class of compoundspertains to ABX3 family where A stands for Cs or Rb, B for a magnetic ion, eitherMn, Cu, Ni, or Co, and X for one of the halogens, Cl, Br, or I. Depending on thekind of anisotropy in the compound relevant model changes. For those with a strongeasy-axis anisotropy Ising model is adapted: this is the case of CsNiCl3, CsNiBr3, andCsMnI3. Easy-plane anisotropy as present in CsMnBr3 and CsVBr3 leads to a descrip-tion with XY model. As for systems with a very weak anisotropy, such as CsVBr3

and RbNiCl3, they let Heisenberg model correctly describe them. Another reason whystacked systems have been so widely studied is their amenability to mean-field analysis[97, 99, 101, 100]. As already discussed a twenty-year-long controversy opposed propo-nents of a continuous transition from the paramagnetic to the ordered phase in XY andin Heisenberg stacked triangular antiferromagnet to opponents claiming these transi-tions were first-order. After the non-perturbative renormalisation group approach byTissier and coworkers [116, 118, 117], and various numerical treatment, Ngo and Diepproposed clear numerical evidence of a first-order transition thanks to Wang-Landauflat histogram algorithm implemented on quite large clusters [89, 88] (and referencestherein for previous numerical works).

At the other end research on 2dmodels and quasi-2d compounds has generated fewerpublications, many extensively discussing zero-field behaviour. The work by Miyashitaand Shiba on the one hand [85] and by Lee and coworkers on the other hand [66, 67]really started the investigations on 2d XY model which were complemented by Kawa-mura’s spin-wave calculations [46] and Korshunov’s extensive analysis of symmetry andtopological excitations [57, 56, 60, 59]. The interest rose due to the emergent chiral tran-sition as already forecast by Villain [122]. As a consequence two symmetry-breakingoccur, which at zero-field induces two distinct transitions that some authors failed todistinguish [66, 67] as opposed to others [85, 68, 134, 69, 91]. Yet finite-field behaviourhas lacked thorough careful investigations despite the quite fine discussion proposedby Korshunov [57], which was a motivation to undertake such investigations. As faras Heisenberg triangular antiferromagnet is concerned if its quantum version has beenrather popular the classical model has been only quite partially studied. Zero-field be-haviour, after the pioneering and inspiring topological analysis proposed by Kawamuraand Miyashita [51, 50], did attract some interest [5, 6, 109, 133, 132, 53]. As this modelhas a continuous symmetry in zero-field, after Mermin-Wagner theorem it is expectedno finite-temperature transition can occur. Kawamura and Miyashita challenged thisview putting forward the existence of two regimes with distinct properties of the stablepoint topological defects, namely Z2 vortices. Further numerical achievements [53] and

1.3. TRIANGULAR ANTIFERROMAGNET: HISTORICAL PERSPECTIVE 11

an experimental realisation of an Heisenberg triangular antiferromagnet [87] proposeconvincing support for this transition. Finite-field behaviour contrastingly has been atbest overlooked [52]. With so little theoretical work on it and with new experimentalresults on quasi-2d compounds (Rb4Mn(MoO4)3, Nakatsuji et al., private communica-tion) did require a correct study.

In this context the work presented in this dissertation intends to propose a clearpicture on the whole phase diagram of classical Heisenberg model (Chap. 2) and itsanisotropic variants, namely easy-axis (Chap. 3) and easy-plane anisotropic models(Chap. 4) based on a blend of numerical simulations and symmetry analysis withoutany heavy technical apparatus. If studies were published presenting elements of a phasediagram for anisotropic models [84, 83], the anisotropy was exchange anisotropy. Froman experimental point of view single-ion anisotropy as used here seems more relevant.Furthermore from a theoretical point of view its perturbing impact is much more dra-matic. As easy-plane anisotropic Heisenberg triangular antiferromagnet is presentedand since a precise phase diagram was still absent, XY triangular antiferromagnet hasbeen studied as well and results are presented in Chap. 4.

12 CHAPTER 1. INTRODUCTION

Chapter 2

Heisenberg TriangularAntiferromagnet

In this chapter we deal with the isotropic version of Heisenberg triangular antiferro-magnet (HTAF); in a way this is the original version that is referred to when speakingof Heisenberg antiferromagnetic model on the triangular lattice. In the first section themodel is briefly introduced and key properties of its Hamiltonian are exposed. Thenresults on zero-field behaviour are reviewed and critically compared. Sec. 2.3 and 2.4are devoted to finite-field behaviour: first we discuss symmetries of HTAF in an exter-nal field, then our numerical calculations for this system are presented and the phasediagram of HTAF proposed. Considering existing results on zero-field behaviour ofHTAF, a new study has seemed worthless; yet to support this viewpoint and to drawa complete picture of the phase diagram of HTAF as here intended, a critical review ofliterature on this topic is included.

2.1 Model

To deal with electronic crystalline systems it is necessary to take into account twocompeting energies describing the behaviour of electrons: Coulomb repulsion and ki-netic energy (hopping). If the incompletely filled orbitals (3d in iron-group elementsand 4f in rare-earth elements) are decribed by localised orbitals, this is encompassedin so-called Hubbard model:

H =∑

n,n′,s

bn′−na†n′,san,s + U

∑

n

a†n,↑an,↑a†n,↓an,↓ (2.1)

where n indexes sites and s is a spin index, a†n′,s a fermionic creation operator creatingan electron at site n with spin s and an,s the associated annihilation operator.

13

14 CHAPTER 2. HTAF

This Hamiltonian can be written in another form when developing it in the low-energy sector:

H = J∑

〈i,j〉Si · Sj (2.2)

where I have assumed for simplicity symmetry in the bn′−n terms and then a singleexchange energy (J can indeed be written in terms of an exchange integral). ThisHamiltonian is called Heisenberg Hamiltonian [136]. When considering systems withlarge spins it is quite appropriate to describe spins as classical vectors which makesHeisenberg model more tractable. In systems for which this work is relevant such anapproximation is correct (most of them are rare-earth compounds) and from now ononly classical models are used unless otherwise stated. For simplicity classical spins arenormalised to one.

Hereafter we consider HTAF with an applied field:

H =∑

〈i,j〉Si · Sj − h ·

∑

i

Si (2.3)

written in the units of the exchange constant J . Even though we did not specificallywork on zero-field behaviour a glimpse on what happens without any field will be givenfor the phase diagram to be entirely discussed.

Let’s discuss the ground states of (2.3) first at zero field then at finite field. Apreliminary observation is the following transformation of (2.3):

H =12

∑

∆

(S∆,1 · S∆,2 + S∆,2 · S∆,3 + S∆,3 · S∆,1) +h6

∑

∆,i

S∆,i (2.4)

=14

∑

∆

ÇS∆,1 + S∆,2 + S∆,3 −

h3

å2

+ const. (2.5)

where summation runs over triangular plaquettes ∆.This transformation makes it plain that to minimise the Hamiltonian a sufficient

condition is to equate each square in the sum to zero, which is possible: it even makesit obvious that imposing a specific structure on a given triangle imposes the config-uration on the whole lattice. Let’s indeed consider a given triangle ∆: let’s pick upone configuration satisfying S∆,1 + S∆,2 + S∆,3 = 0 (infinity of solutions, among whichthose with spins pointing at 120 degrees from one another equally share frustrationon the three bonds); then on a neighbouring triangle ∆′ there is a unique solution toS∆′,1 +S∆′,2 +S∆′,3 = 0 as the two spins pertaining to both ∆ and ∆′ are already fixed,qed. As a consequence ground states respect a three-sublattice pattern, the so-called√

3×√

3 pattern that is associated with ordering wave vector q0 = (4π/3, 0). Obviouslythe condition is a necessary one as well: if one of the squares in the sum were non-zerothen the energy of the considered configuration would be strictly larger than the oneof configurations with all squares equal to zero – these configurations exist as already

2.2. BRIEF REVIEW OF ZERO-FIELD BEHAVIOUR 15

demonstrated – and hence would not be a ground state, qed. Such a constraint how-ever lets a continuous degeneracy appear in the system. There are indeed six degreesof freedom corresponding to the three unit-length vectors (spins) constrained by threeequations: three free parameters remain. A closer look at this extra degeneracy showsthat it corresponds to the degrees of freedom of the three-spin structure on a triangularplaquette, thence the corresponding order parameter space SO(3).

As previously discussed the ordering respects a three-sublattice pattern: it is thusnatural to describe the spin structure as:

〈Si〉 = l1 cos(q0 · ri) + l2 sin(q0 · ri) + m (2.6)

where m is static magnetisation, which is zero at zero field, and l1 and l2 are antiferro-magnetic ordering vectors. In this language, 120-degree structure corresponds to a pairof orthogonal vectors l1 ⊥ l2, |l1| = |l2| and m = 0. In case of a distorted 120-degreestructure (either due to field or easy-axis anisotropy) m 6= 0. Let’s present two otherconfigurations of importance as well (an overall view of low-field planar configurationscan be found in Fig. 3.1). Collinear fully-ordered configuration, so-called up-up-downconfiguration, is characterised by l2 = 0 and l1 and m collinear. High-field planar V-shape configuration is defined by l2 = 0, l1 having components both along the field andtransverse to it, and m along the applied field direction.

2.2 Brief review of zero-field behaviour

At zero field this extra continuous degeneracy of ground state manifold coincideswith order parameter space which is SO(3) the group of rotations in 3 dimensions, or ina more pictorial language the rigid body of three spins in a given triangle. Let’s discusspossible phase transitions in such an order parameter space. First, an important remarkshould be put forward: in this two-dimensional system there cannot be any finite-temperature phase transition associated with the breaking of a continuous symmetry,after Mermin-Wagner theorem [77]. With this viewpoint there shouldn’t be any finite-temperature transition in HTAF. However the conclusion becomes less obvious whenthe question is dealt with from a topological viewpoint: indeed stable topological defectsexist in HTAF, which may induce a phase transition in a similar way to what happensin XY antiferromagnet as described by Berezinskii [10, 11] on the one hand and byKosterlitz and Thouless [61] on the other hand (see Sec. 1.1 and 4.1) with the so-called BKT transition. Such a claim was made by Kawamura and Miyashita [51, 50].Topologically stable defects are given by homotopy groups [82, 76, 75, 79]: line defectsby zeroth homotopy group, point defects by first homotopy group and instanton bysecond homotopy groups. In this case the single non-trivial homotopy group is the firstone: π1(SO(3)) = Z2, which implies the system admits stable Z2 vortices. How can bothviewpoints be reconciled? If later studies based on renormalisation group applied to a

16 CHAPTER 2. HTAF

relevant non-linear σ model [5, 6] predict a zero-temperature phase transition pertainingto O(4) universality class, further investigations either based on harmonic expansionand analytical predictions [133] or on Monte Carlo techniques [109, 132, 53] indicate tworegimes exist: a low-temperature one, T < T ∗, which is consistent with renormalisationgroup predictions, and a high-temperature one, T > T ∗ in which the influence offree vortices has to be taken into account, which is not done in renormalisation groupanalysis. Wintel et al. and Southern and Young thus point out a crossover between tworegimes at T = T ∗, T ∗ ≈ 0.28 (Kawamura et al. [53] find T ∗ = 0.285± 0.005): in low-temperature regime spin correlation length and antiferromagnetic susceptibility are welldescribed by renormalisation group; in high-temperature regime a fit to BKT behaviouris much more accurate. The latter indicates a certain similarity with BKT transition.Their conclusion relies on the study of spin correlation length ξ, susceptibility χ(q0)and spin stiffness. ξ is estimated with Orstein-Zernicke relation:

χ(q) =1L2

∑

i,j

〈Si · Sj〉Teiq(Ri−Rj)

=χ(q0)

1 + ξ2(q − q0)2(2.7)

The temperature T ∗ of this sharp crossover is compatible with Miyashita’s and Kawa-mura’s results who found with quite a simple Monte Carlo approach T ≈ 0.3 as atransition temperature. Contrary to above cited authors who consider it merely as asharp crossover, Kawamura supports the idea of a peculiar transition. The original ar-gument by Kawamura and Miyashita [50] is based on a thorough discussion of vorticesin HTAF. As said these are Z2 vortices and not Z vortices as is the case in XY systemsthat exhibit BKT transition. To track the behaviour of vortices they introduce a vor-ticity function that is defined on the dual lattice links: 120-degree structure now standsat a vextex of the dual lattice; on the oriented links in the dual lattice the rotation of120-degree structures from one end of the link to the other is a well-defined object thatcan be represented by an SU(2) matrix U . Vorticity V is the function defined on closedloops C as follows:

V (C) =12tr

(

∏

i∈CUi

)

(2.8)

They showed that vorticity undergoes a transition between a perimeter-law asymptoticbehaviour at low-temperature (T < T ∗) and an area-law asymptotic behaviour at hightemperature (T > T ∗), which is similar to the behaviour of Wilson’s loops [131, 55]:

VR = 〈V (CR)〉 −→R→∞

®exp(−αA) T > T ∗

exp(−βR) T < T ∗(2.9)

where CR is a loop of length R, A is the enclosed area. In fact this point emphasisesthe role played by free vortices to drive the system from low-temperature phase to

2.2. BRIEF REVIEW OF ZERO-FIELD BEHAVIOUR 17

high-temperature phase. On this particular point all groups do agree. On the natureof the change at T ∗ there is a disagreement. Kawamura and coworkers further supporttheir viewpoint [53] pointing out the existence of two different length scales. Indeedas already discussed in their early paper [51, 50], the transition cannot be a BKTone. It is then hardly surprising that they disagree with other groups like Wintel andhis coworkers who are specifically looking for a BKT transition. Let’s further developthe argument. Vorticity exhibits a sharp drop from low-temperature regime to high-temperature regime, similarly to what happens in a usual BKT transition. There ishowever an important difference: in HTAF spin stiffness doesn’t undergo any such sharpdrop whereas in usual BKT transition both spin stiffness and vorticity behave the sameway with a jump at the same temperature. Here comes the main difference between aBKT transition and what occurs in HTAF: the existence of two different length scales,namely the one of spinwaves and the one of vortices [53]. The former doesn’t diverge atthe transition contrary to the latter. In other terms the system is characterised by twodifferent stiffnesses. Spin correlation function C(rij) = 〈Si · Sj〉 can be factorised intoa spinwave contribution and a vortex contribution, the same way it is done for BKTtransitions: C(rij) = Csw(rij)Cv(rij) [40]. Assuming the normal exponential form forcorrelation functions, correlation length then reads:

ξ =ξswξvξsw + ξv

(2.10)

Consequently in the vicinity of the transition where ξv ≫ ξsw correlation length exhibitsa weak essential singularity:

ξ ∼ ξswÇ

1− ξswξv

å(2.11)

Spin correlation length remains finite at low temperature: spin correlation decays expo-nentially both above and below the transition. This exponential decay was expected dueto Mermin-Wagner theorem; a more sophisticated argument based on topological con-siderations consists in the following: symmetries involved in a phase transition can befound and analysed through the space associated with low-temperature phase removingits topological defects, here vortices, and retaining all other parameters equal. In topo-logical language this means calculating the universal covering of the order parameterspace. Here the universal covering of SO(3) is S3, the three-dimensional sphere. S3

is the order parameter space of Heisenberg ferromagnetic four-component spin systemas well: in two dimensions, spin correlations in this system decay exponentially, whichindicates the same occurs in HTAF at zero field. Low-temperature phase is not anordered state in the traditional way but is topologically ordered as the single accessi-ble sector is the one without any free vortex. The nature of low-temperature phase isthus rather uncommon: Kawamura and coworkers proposed to call such a state that isneither a liquid nor an ordered AF state a spin gel [53].

18 CHAPTER 2. HTAF

2.3 Finite-field behaviour: symmetry discussion

As discussed in the preceding section, zero-field behaviour excludes finite-temperaturetransition to a long-range ordered phase with a gapped spectrum, ie a massive phase.It was consequently quite surprising to realise so far published phase diagrams depicttransition lines going to a finite-temperature multicritical point at zero-field [52], whichdisregards an essential feature of HTAF, namely that zero-field configurations are mass-less. Indeed the existence of such a multicritical point would imply an infinitesimallysmall field makes the system massive: this is quite questionable a statement! For thisreason it was necessary to examine anew finite-field behaviour in Heisenberg triangularantiferromagnet. A more natural viewpoint is indeed that finite-field transition linesclose at (T, h) = (0, 0).

Looking back at the antiferromagnetic order parameter in Eq. (2.6) symmetries atfinite field can be discussed. If we consider a simple translation Ta with a lattice vectora (ri → ri + a) the order parameter is transformed as

Ta[l1 + il2] = (l1 + il2)e−iq0·a (2.12)

with the phase factor q0 · a ∈ 0,±2π/3, which means that an inherent discrete Z3

symmetry exists besides the continuous S1 symmetry associated with free rotationsabout the direction of the applied field z. Hence the system is governed by a compoundsymmetry S1 ⊗ Z3. Yet as seen with (2.5) a continuous degeneracy still exists: itdoesn’t match any symmetry in the Hamiltonian. In such a case it is legitimate towonder whether this degeneracy of ground state manifold is robust against fluctuations.For a degeneracy not associated with any symmetry of the Hamiltonian, no Goldstonemode exists [34], which implies there is no entropy gain for such states. In our caserelevant fluctuations are thermal ones. As shown by Sheng and Henley [106], thermalfluctuations must be calculated at higher order than the second one for degeneracy tobe removed; a selection does occur reducing degeneracy to S1 ⊗ Z3. This reduction ofdegeneracy is an example of an ‘order by disorder’ scheme [124].

At low temperature and low field the expected stable configuration is the so-called120-degree configuration which is a natural solution of (2.5). A simple calculationshows that 120-degree configuration is not stable above h = 3. Since ground stateconfigurations are defined by the configuration on any triangle, let’s write (2.3) usinga three-sublattice pattern:

H =N

3× 1

2× 6 (m1 ·m2 + m2 ·m3 + m3 ·m1)−

N

3h (m1 + m2 + m3)

= Nε

where mi = 〈Si〉 is thermal averaged value of the spins of i−th sublattice, N is thenumber of spins, and ε the average energy per spin:

ε = m1 ·m2 + m2 ·m3 + m3 ·m1 −h3

(m1 + m2 + m3) (2.13)

2.3. FINITE-FIELD BEHAVIOUR: SYMMETRY DISCUSSION 19

θ

Figure 2.1: Distorted 120-degree structure with the definition of the angle θ. The structureis rotationally symmetric about the dashed axis that stands for the external field direction orthe anisotropy axis (as discussed in Sec. 3.1). The three arrows stand for the three spins of atriangular plaquette.

The energy of a deformed 120-degree structure (see Fig. 2.1) then reads:

ε = cos 2θ − 2 cos θ +h

3(1− 2 cos θ) (2.14)

The minimisation of (2.14) with respect to θ yields:

cos θ =12

Ç1 +h

3

å(2.15)

which shows such a configuration exists if and only if h 6 3.At T = 0 a transition must therefore occur between low-field (h < 3) and high-field

(h > 3) regimes. In fact an intermediate phase separates the low-field low-temperatureand quasi long-range ordered 120-degree phase from high-field configuration [35]. Theintermediate phase corresponds to one-third magnetisation plateau: it is an up-up-downcollinear configuration. Consequently spin-flop transition doesn’t exist: instead doublecontinuous transitions occur. At T = 0, this intermediate collinear phase is confined tothe point h = 3 that hence is a critical point that can be studied in quantum limit asan interesting quantum critical point. At finite temperature, thermal fluctuations arestrong enough to stabilise this collinear phase in a larger domain of temperature-fieldplane in low field region (high fields obviously suppress up-up-down structure). At highfield thermal fluctuations select a planar configuration instead of umbrella structure[106]: in this configuration two sublattices develop the same magnetisation [22] andnon-zero transverse magnetisation exists as illustrated in Fig. 2.2.

20 CHAPTER 2. HTAF

βα

Figure 2.2: High-field planar structure that is stabilised at h > 3, referred to as V-shapeconfiguration in the text. It is characterised by l2 = 0, l1 with components both along thedirection of the dashed line that is the direction of field and of the easy-axis in case of easy-axisHTAF, and transverse to this direction and m along the dashed line. The three arrows standfor the three spins of a triangular plaquette.

The natural question is then the nature of phase transitions occurring in the system.Such a treatment can be carried out discussing the order parameter symmetries. Con-sidering the order parameter space, S1⊗Z3, a discrete and a continuous symmetry haveto be broken. Different scenarios are possible as Korshunov discussed it [60, 57, 58, 59].Naturally there are three of them: either (i) the restoration of S1 occurs before therestoration of the discrete symmetry or (ii) it is the opposite way round or (iii) thereis a single transition that belongs to a new universality class. The latter case seemsquite exotic and rather unlikely: the reason why it may have been suggested in somenumerical studies is that numerical accuracy at that time was too limited to distin-guish both transitions. Modern numerical investigations have ruled it out. First twoscenarios involve excitations associated with each symmetry: vortices for the continuoussymmetry, domain walls for the discrete symmetry. As a consequence it is possible toinvestigate the restoration of the compound symmetry analysing the behaviour of theseexcitations, or in other words tools used to investigate each transition can be used (spinstiffness and/or vorticity for S1, correlation function and/or susceptibilities for Z3). Amajor difference between both scenarios is that in the second case fractional vorticesform at kinks on domain walls [60, 57, 58], which implies the jump in spin stiffness islarger than it is in case of integer vortices, namely (2q2/π)T for 1/q vortices. This canbe considered a definitive way to distinguish between both scenarios.

To discuss the succession of phase transition it is necessary to distinguish low-fieldand high-field regions as the involved spin configurations are different. In low-fieldregion, as previously mentioned, thermal fluctuations stabilise up-up-down collinearphase, which thus appear as an intermediate phase between 120-degree configurationand both high-field configuration and high-temperature (paramagnetic) phase. Thismeans that discrete symmetry-breaking occurs first (in this case Z3) then continu-ous symmetry-breaking (S1). Indeed collinear structure only breaks Z3 symmetry andleaves S1 symmetry unbroken. The latter breaks down in 120-degree configuration.Furthermore leading order parameters associated with each phase transition are dif-

2.3. FINITE-FIELD BEHAVIOUR: SYMMETRY DISCUSSION 21

ferent: Z3 symmetry-breaking is associated with the component along field directionwhereas S1 symmetry-breaking is associated with transverse component. As orderingoccurs along different components in low-field region, the order of phase transition isclear and associated universality classes as well: the upper transition corresponds to theordering along the field (up-up-down structure) and pertains to three-state clock modeluniversality class (cf.[9]) whereas the lower transition corresponds to the ordering alonga transverse direction (120-degree structure) and pertains to a BKT universality class.

At high fields the situation is less clear because there is only one ordered phasewhich corresponds to a planar spin configuration with spins on two sublattices iden-tical and with a finite transverse magnetisation, so-called V-shape configuration (seeFig. 2.2). As l2 = 0 in this high-field configuration, there is a single order parameterinvolved in the breaking of two symmetries. In real space the ordering actually occursalong a single component, namely the transverse one; the ordering of parallel compo-nent is induced by the ordering of transverse component as can be seen considering thefollowing term of Landau-Ginzburg functional that comes into play: Szq

ÄS⊥qä2

, whichindicates that transverse component acts as an ordering field on parallel component.As previously discussed the breaking of compound symmetry Z3 ⊗ S1 at once is quiteunlikely: successive transitions corresponding to the successive breaking of each com-ponent of this compound symmetry are thus expected. The tricky question to answerto is then the order of these transitions. Korshunov demonstrates case (ii) implies theexistence of fractional vortices centered at kinks on domain walls. Numerically it canbe checked examining spin stiffness jump at binding-unbinding transition: the jumpfor a 1/q fractional vortex is indeed (2q2/π)TBKT . From this observation he drawsthe conclusion that scenario (i) necessarily occurs: the transition temperature for thebinding-unbinding transition of fractional vortices is indeed much lower than the es-timate for the transition associated with domain walls (percolation threshold), whichis contradictory with scenario (ii). Consequently there is no fractional vortices andscenario (i) should occur. In next section a way to numerically find out the transitionsequence is proposed.

22 CHAPTER 2. HTAF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5temperature T

0

1

2

3

4

5

6

7

8

9

fiel

d h

B

C

A

DD’

Figure 2.3: Phase diagram of Heisenberg triangular antiferromagnet. Solid lines are a guideto the eye linking calculated points. Dashed lines indicate theoretically expected transitionsfor which precise numerical results have not yet been obtained. Spin structures are sketchedby the configuration on a triangular plaquette with z axis assumed vertical and xy planeperpendicular to the plane of the sheet.

2.4. FINITE-FIELD BEHAVIOUR: NUMERICAL DETERMINATION 23

2.4 Finite-field behaviour: numerical determination

The phase diagram is numerically calculated using a hybrid algorithm that asso-ciates an optimised single-spin Monte Carlo algorithm with over-relaxation process (seeApp. E). The latter increases the portion of explored configuration space and thereforereduces the risk to remain trapped in a local minimum. With single-spin algorithm therisk is indeed quite important to get stuck in a local minimum, which hinders ergodicityproperty, an essential feature for Monte Carlo algorithm to be right. Our simulationshave been carried out on finite-size rhombic clusters with periodic boundary conditionsand linear sizes 24 6 L 6 120 compatible with a three-sublattice pattern in order toavoid artificial frustration effects.

2.4.1 Preliminaries

Numerical calculation of a phase diagram consists in the identification of the rele-vant order parameters in the system according to symmetries at stake in the variousphase transitions and the calculation of the most accurate numerical estimators of theseorder parameters or of observables reflecting them. Then it is possible to locate singu-larities in temperature or field dependence of these estimators, to use finite-size scalingto extract information about the precise location of phase transition and its nature(via the calculation of critical exponents). Both field and temperature sweeps havebeen carried out. In case of temperature sweeps the initial configuration is a randomconfiguration in the paramagnetic region of phase diagram that is cooled down untilthe temperature range of interest. For field sweeps the initial configuration is a fullypolarised configuration in the saturated phase.

Beside observables that are explicitly discussed and presented in the Monte Carlonumerical parts of this work, few calculations of the ordering vectors l1 and l2 have beencarried out in terms of the third-order invariant (lα1 + ilα2 )3. This approach is yet moreadapted to mean-field analysis and within a Monte Carlo investigation the calculationof more directly accessible observables is more interesting, which is the reason why Ionly present results for these observables.

Let’s discuss how this general program is here implemented. Of course one of themost straightforward idea is to track specific heat which can be defined as a measure-ment of energy variance:

c =〈E2〉 − 〈E〉2NsT 2

(2.16)

where E is the internal energy and Ns = L2 the number of spins in the consideredcluster. Specific heat is however hardly suited to precisely locate such transitions asBKT (for the appearance of specific heat peak at a higher temperature T than TBKT , seeeg Fig. 9.4.3 in [20]) and often requires a fine sampling in the vicinity of phase transitionto extract a precise estimate of transition temperature. However specific heat can be

24 CHAPTER 2. HTAF

helpful to identify the nature of a phase transition thanks to the determination of criticalexponent α. Numerical estimators associated with other observables have therefore beenconsidered as well: spin structure factor components; Binder’s cumulant correspondingto spin structure factor; spin susceptibility; spin stiffness. Let’s detail these differentquantities. Spin structure factor is the Fourier transform of spin configuration at theordering wave vector q = q0 = (4π/3, 0) and its components constitute relevant orderparameters in this problem. In our numerical calculations we have calculated variouseven powers of its components; for example:

Ämαqä2

=1N2s

∑

i,j

〈Sαi Sαj 〉eiq·(ri−rj) (2.17)

A frequently used quantity to locate phase transitions (first and second order andinfinite order as well) is Binder’s cumulant that is a fourth-order quantity defined asfollows:

UA =〈A4〉〈A2〉2 (2.18)

where A is the order parameter of interest. At critical points Binder’s cumulant becomessize-independent: as a consequence plots of Binder’s cumulant versus temperature fordifferent cluster sizes cross at a second-order phase transition, merge at a BKT transi-tion (plots are on top of each other in low-temperature phase). In this case finite-sizescaling is straightforward. As a fourth-order quantity Binder’s cumulant is a statisticalestimator that exhibits fluctuations that may be large especially in frustrated systems.A remedy is to consider a second-order quantity with similar properties: a ratio of spin-spin correlation functions fulfilling these requirements has been proposed by Tomita andOkabe [120, 112] (see App. D):

Γy =g(L/2, L, T )g(L/4, L, T )

(2.19)

where g(r, L, T ) = L−d∑

i〈Si · Si+r〉 is the spin-spin correlation function calculated ona cluster of linear size L, at temperature T for spins at distance r apart (i + r shortfor neighbours at distance r of spin i). This ratio often exhibits better statistics thanBinder’s cumulant. As both observables are independent of any assumption on thenature of the transition (order, class of universality) they are quite useful a referenceto check hypotheses made for the scaling of other parameters such as susceptibility.

Susceptibility is indeed a key element of various finite size scaling techniques tolocate transitions and to calculate critical exponents, details of which are developedin App. D. Furthermore spin susceptibility is a second order quantity, which impliesweaker fluctuations than for Binder’s cumulant. Spin susceptibility characterises linearresponse of spins to a magnetic field. It is a second-order tensor the indexes of which

2.4. FINITE-FIELD BEHAVIOUR: NUMERICAL DETERMINATION 25

run over field directions in spin space. As there are here no off-diagonal terms, it thenreads:

χαq =1Ns

∑

i,j

〈Sαi Sαj 〉Teiq·(ri−rj) (2.20)

The most widely used means of studying BKT transition remains finite size scalingof spin stiffness ρS, which is associated with the very origin of this kind of transition,namely the unbinding of stable point defect, vortices. At BKT transition spin stiffnessexhibits a jump the height of which is universal: ρS = 2TBKT/π in case of integervortices (ρS = 0 in high-temperature phase). Spin stiffness is defined as a generalelasticity coefficient in response to a weak nonuniform twist of spins δφα(r) performedabout a certain direction α in spin space. In our case it is sufficient to consider twistsabout z direction, the direction of applied field, in spin space while all directions inlattice plane are equivalent thanks to sixfold rotational symmetry. A single parameteris thus left

δF =ρS2

∫

d2r[∇φz(r)]2 (2.21)

Choosing a twist with a uniform gradient along an arbitrary direction e in the latticeplane, one obtains in spherical coordinates

Si · Sj = cos(θi) cos(θj) + sin(θi) sin(θj) cos(ϕi − ϕj) (2.22)

with ϕi = ϕi + δφ e · ri.Calculating the change in the free energy up to second order in a small δφ and

normalizing the result per unit area one obtains [92, 115, 127]:

ρS = − 1L√

3

∑

〈i,j〉〈Sxi Sxj + Syi S

yj 〉+

2LT√

3

≠ß∑

〈i,j〉(Sxi S

yj −Syi Sxj )[e · (ri − rj)]

™2∑(2.23)

2.4.2 Numerical results

As discussed in section 2.3, low-field and high-field regions have to be treated indifferent ways as the stabilised structures are different. Let’s start with low-field re-gion: collinear ordering and the ordering into 120-degree configuration are expected.The former happens along field direction z and can thus be probed plotting Binder’scumulant associated with z-component of S(q) versus temperature:

U zq =〈(Szq )4〉〈(Szq )2〉2 (2.24)

The crossing of plots for different cluster sizes indicates the transition into a long-rangeordered phase. As HTAF adopts a planar ordering and up-up-down collinear structure

26 CHAPTER 2. HTAF

0.321 0.322 0.323 0.324 0.325 0.326 0.327 0.328 0.329 0.33temperature T

0.5

0.55

0.6

0.65

0.7

0.75

0.8sc

aled

spi

n su

scep

tibili

ty χ

qz / L

2-η L = 48

L = 60L = 72L = 96

η = 4/15

0.321 0.322 0.323 0.324 0.325 0.326 0.327 0.328 0.329 0.33temperature T

1

1.05

1.1

1.15

1.2

1.25

1.3

Bin

der

cum

ulan

t U

qz

L = 48L = 60L = 72L = 96

Figure 2.4: Scaling of spin susceptibility χzq with critical exponent η = 4/15 correspondingto three-state clock-model universality class as χzq/L

2−η yields a transition temperature Tc =0.327 ± 0.001 (left). The temperature obtained with Binder’s cumulant U zq as shown on theright is the same. Both figures are drawn at field h = 1.0.

breaks Z3 symmetry the phase transition should pertain to three-state clock modeluniversality class as is the case for XY model. With this viewpoint it is then possibleto scale z-component of spin susceptibility (calculated as in (2.20) with α = z) with thecorresponding critical exponent η, η = 4/15 in this case. The transition temperatureis obtained as the temperature of the crossing point of plots for different cluster sizes.Fig. 2.4 illustrates both methods at h = 1.0.

As the system is cooled down a BKT transition is expected for transverse spin com-ponent. This transition can be located using transverse spin susceptibility with η = 1/4for its scaling as shown in Fig. 2.5. This method can be used both during temperatureand field sweeps: given the shape of the transition line BDD′ into 120-degree phase itis indeed convenient to sweep field at fixed temperature in low-temperature region —the closer to straight angle is the angle between the sweeping line and the transitionline, the sharper the location of the transition. Fig. 2.6 illustrates such a field sweepat T = 0.06 with both transverse spin susceptibility and Binder’s cumulant; insets pro-vide an enlarged view around the transitions. Defined in a natural way using (2.20),transverse spin susceptibility reads:

χ⊥q =1N2s

∑

i,j

〈Sxi Sxj + Syi Syj 〉

Teiq·(ri−rj) (2.25)

In the final diagram (see Fig. 2.3) transition lines D′D at very low field appears asdashed lines: this expresses the difficulty to precisely locate transitions in the vicinityof zero-field. Even though hints of a transition can be sensed, statistical errors dueto large fluctuations in this critical region (due to the nature of low-temperature zero-field phase, a severe slowing down of the convergence of Monte Carlo algorithm occurs)prevent us to fix for sure transition lines. However the argument earlier put forward

2.4. FINITE-FIELD BEHAVIOUR: NUMERICAL DETERMINATION 27

0.265 0.27 0.275 0.28 0.285temperature T

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

scal

ed s

pin

susc

eptib

ility

χqt /

L2-

η L = 48L = 60L = 72L = 96L = 120

η = 1/4

Figure 2.5: Scaling of spin susceptibility χ⊥q with critical exponent η = 1/4 corresponding

to BKT universality class as χ⊥q /L2−η at h = 1.0.

(an infinitesimal field cannot turn a massless phase into a massive one) lets us interpretthese numerical hints as a further support for our proposition that transition lines goto (T, h) = (0, 0).

Above h = 3.0 ordering occurs along transverse component. First a transition thatcan be easily investigated is the transition between up-up-down collinear phase andhigh-field phase: as previously explained there is no spin-flop first-order transition buta continuous transition. From symmetry analysis this transition is associated with S1:it can thus be located the same way as the transition between collinear phase and120-degree phase as can readily be seen in Fig. 2.6.

Concerning the ordering process from paramagnetic phase or saturated phase, thedelicate problem of the breaking of compound Z3 ⊗ S1 symmetry for a single order-parameter, namely

ÄS⊥qä2

, has to be dealt with. As stated among three possible sce-narios one is expected to occur: first the breaking of discrete Z3 symmetry, then thebreaking of continuous S1 symmetry. To find out whether numerical results supportor not this hypothesis it is necessary to extract the relevant information from data.Observables giving access to each symmetry separately have to be considered. As faras Z3 is concerned, scaling of transverse spin susceptibility can be used with criticalexponent η = 4/15 corresponding to the class of universality of three-state clock model.This method yields a transition that is confirmed by the study of both the ratio ofcorrelation functions Γ⊥ and Binder’s cumulant U⊥q as illustrated in Fig. 2.7. As forBKT transition spin stiffness ρS is used. It is plotted versus temperature; the heightof the jump in this case is incompatible with fractional vortices: integer vortices are at

28 CHAPTER 2. HTAF

stake. The temperature at the intersection of ρS(T, L) with the line (2/π)T is calledTL. The extrapolation to thermodynamic limit reads (cf App. D and Eq. D.21)

TL − TBKTTBKT

=1

2(lnL+ c)(2.26)

This requires a non-linear fitting with two parameters, TBKT and c = − lnL0. Such afit is illustrated in Fig. 2.7. In this way BKT transition is obtained at slightly lowertemperature than the one associated with discrete symmetry breaking. These resultssupport Korshunov’s defense of scenario (i). Interestingly that in the case of isotropicHTAF the splitting between transitions is much tinier than in the case of easy-axisHTAF as can be seen in Fig. 3.8

2.4. FINITE-FIELD BEHAVIOUR: NUMERICAL DETERMINATION 29

2.4 2.5 2.6 2.7 2.8 2.9 3field h

0

0.5

1

1.5

2

scal

ed s

pin

susc

eptib

ility

χqt /

L2-

η

L = 48L = 60L = 84L = 96

2.5 2.550.8

1

1.2

3.051

1.2

1.4

1.6

1.8

η = 1/4

2.4 2.6 2.8 3 3.2field h

1

1.2

1.4

1.6

1.8

Bin

der

cum

ulan

t U

qt

L = 48L = 60L = 72L = 96

2.5 2.551

1.05

1.1

L = 48L = 60L = 72L = 96

3 3.11

1.1

1.2

L = 48L = 60L = 72L = 96

Figure 2.6: Field sweeping at T = 0.06. Scaling of spin susceptibility χ⊥q with critical

exponent η = 1/4 corresponding to BKT universality class as χ⊥q /L2−η (top). Insets give an

enlarged view of crossing points that corresponds to BKT transitions: h 6 2.50 correspondsto 120-degree phase, 2.50 6 h 6 3.04 to collinear phase, and 3.04 6 h to high-field phase.Binder’s cumulant plotted versus temperature (bottom) confirms these transitions; insets offeran enlarged view on transition regions.

30 CHAPTER 2. HTAF

0,17 0,18 0,19 0,2 0,21 0,22 0,23

temperature T

0,5

1

1,5

2

2,5

scal

ed s

usce

ptib

ility

χqt /

L2-

η

L = 48L = 96L = 132L = 192

η = 4/15

0.14 0.16 0.18 0.2 0.22temperature T

1

1.05

1.1

1.15

Bin

der

cum

ulan

t U

qt

L = 48L = 96L = 132L = 192

0.14 0.16 0.18 0.2 0.22temperature T

0.8

0.85

0.9

0.95

ratio

of

spin

cor

rela

tion

func

tions

Γt

L = 48L = 96L = 132L = 192

20 30 40 50 60 70 80linear size L

0.23

0.24

0.25

0.26

0.27te

mpe

ratu

re T

L

Figure 2.7: Comprehensive data for the location of transitions at h = 5.0. Scaling of spinsusceptibility χ⊥q with critical exponent η = 4/15 corresponding to the universality class of

three-state clock-model as χ⊥q /L2−η (upper left). Binder’s cumulant exhibits a merging of

curves for different cluster sizes at the upper transition; it illustrates how hard it is to dis-tinguish close transitions involving the same spin component using Binder’s cumulant (upperright). The ratio of correlation functions Γ⊥ sharply indicates the upper transition as a cross-ing for different cluster sizes; yet it also shows that here considered cluster sizes are somewhattoo small to precisely locate BKT transition the presence of which can be seen in the mergingof curves occurring for the largest sizes (lower left). BKT transition is located thanks tothe scaling of TL the temperature such that ρS(TL, L) = (2/π)TL as described by Eq. D.21:non-linear fitting yields TBKT = 0.203± 0.001 and c = −2.99± 0.001.

Chapter 3

Heisenberg TriangularAntiferromagnet with easy-axissingle-ion anisotropy

In previous chapter a presentation of the phase diagram of HTAF is proposed. Asseen one important characteristic of this diagram is the absence of ordering at zero field.This peculiarity is yet often absent in real compounds due to various perturbation terms;one of particular importance is single-ion anisotropy (see App. A for further details).When considering the easy-axis case as done in this chapter the system retains exactlythe same symmetries as HTAF at finite field provided the field is applied along theanisotropy axis – hereafter I consider the case of a system with a global anisotropyaxis rather than local anisotopy axes. As symmetries at finite field are the same asin HTAF the reader should refer to Sec. 2.3; at zero-field symmetries are specific tothis anisotropic system and are discussed in Sec. 3.1: quite interesting a succession oftransitions occurs, with three transitions (see [73] as well). At a mean-field level thissuccession reflects the change of sign of a sixth-order coefficient in Landau-Ginzburgexpansion of free energy, which is associated with the discrete six-fold symmetry existingin the system. A mean-field analysis was carried out to further discuss symmetriesand for its relevance for quasi-2d layered systems; it ascertains the stability of three-sublattice structures as structures with minimal energy as well. Mean-field treatmentis carried out in Sec. 3.2. Then thermal fluctuations are taken into account through aMonte Carlo procedure and the whole phase diagram is calculated (see Fig. 3.8) bothat zero field (Sec. 3.3) and finite field (Sec. 3.4).

31

32 CHAPTER 3. EASY-AXIS HTAF

3.1 Symmetries at zero field

Let’s first write the Hamiltonian:

H =∑

〈i,j〉Si · Sj − d

∑

i

(Szi )2 − h

∑

i

Szi (3.1)

where terms are expressed in the units of the exchange constant J and spins are classicalspins normalised to 1. In this chapter easy-axis anisotropy is considered, ie d > 0.

In contrast to zero-field spin configuration in Heisenberg triangular antiferromagnet,spin configurations in easy-axis Heisenberg antiferromagnet do exhibit a nonzero staticmagnetisation even at zero field:

〈Si〉 = l1 cos (q · ri) + l2 sin (q · ri) + m (3.2)

where q = q0 = (4π/3, 0). Easy-axis anisotropy orients spin plane perpendicularto xy crystallographic plane and simultaneously distorts the spin structure. Findingdirections and magnitudes of l1 and l2 then becomes quite nontrivial a task.

2zl

,1zl 2xl ,1xl 2zl

l 2zl1z

(a) (c)

(d) (e)

,

(b)

1zl

z

Figure 3.1: Possible three-sublattice planar configurations of the easy-axis triangular anti-ferromagnet. The direction of the easy-axis is indicated by z. Non-zero components of theorder parameter (3.2) are indicated below each configuration.

The same way it was previously discussed (see Sec. 2.3) in order to elucidate sym-metries of different phases, we note that a simple translation by a lattice vector a Ta

(ri → ri + a) transforms the antiferromagnetic order parameter according to

Ta

î(l1 + il2)

ó= (l1 + il2) e−iq0·a , (3.3)

where the phase factor can take only three different values: q0 · a = 0,±2π/3. Hence,besides the group S1 of continuous rotations about z-axis the magnetic structure has an

3.1. SYMMETRIES AT ZERO FIELD 33

inherent discrete symmetry Z3. Such an additional symmetry corresponds to permuta-tions of the three sublattices. In zero magnetic field the time-reversal symmetry impliesinvariance with respect to li → −li, which enlarges Z3 to Z6. The total symmetry groupis, therefore,

G = S1 ⊗ Z6 (3.4)

See also a similar discussion in Ref. [106]. The collinear phases shown in Figs. 3.1(a)-(c) preserve the axial symmetry S1 but break in different ways the discrete symmetrygroup Z6. In terms of the order parameter angle ϕ defined by

l1z = l cosϕ , l2z = l sinϕ , (3.5)

the state in Fig. 3.1(a) corresponds to commensurate values ϕ = 2kπ/6 with an integerk, whereas the configuration in Fig. 3.1(b) has ϕ = (2k + 1)π/6. The third typeof collinear state is described by an arbitrary angle ϕ and is shown schematically inFig. 3.1(c). In such a state the phase ϕ remains unlocked and the sine and cosineharmonics introduced in (3.5) coexist with an arbitrary ratio.

For large enough values of d > dc = 1.5 the magnetic anisotropy induces a highlydegenerate collinear Ising state at zero temperature. Quantum fluctuations can lead,then, to interesting zero- and finite-temperature phases [25, 103]. This collinear orderingresults from the disappearance of deformed 120-degree structures. It can be understoodby a simple calculation of average energy per spin assuming a three-sublattice structurewith 120-degree configuration as shown in Fig. 3.1(d). Let’s introduce mi = 〈Si〉 thethermal average magnetisation of spins in the i−th sublattice. Then the average energyper spin ε reads:

ε = m1 ·m2 +m2 ·m3 +m3 ·m1−d

3

Ä(mz1)2 + (mz2)2 + (mz3)2

ä− h

3(mz1 +mz2 +mz3) (3.6)

The energy of a deformed 120-degree as depicted in Fig. 2.1 at zero field can then beexpressed in terms of the deviation angle θ as

ε = −2 cos θ + cos 2θ − d3

(1 + 2 cos2 θ) (3.7)

The angle θ that minimises ε verifies

cos θ =1

2(1− d/3)(3.8)

which immediately shows the upper bound of d for such a structure to exist, namely dc =3/2. The Ising limit d > dc for interesting it may be is already well-documented andhas not been the focus of my attention. Moderate anisotropy domain which has beenoverlooked for long presents quite interesting properties which are worth the discussion.As a consequence I only develop this case of moderate anisotropy, 0 < d < dc. In Sec. 3.2

34 CHAPTER 3. EASY-AXIS HTAF

d = 1.0 is used. Figures presented in Sec. 3.3 have been calculated with d = 1.0; yetresults for zero-field within Monte Carlo approach have also been obtained for d = 0.2that is the value used to build the whole phase diagram presented in Fig. 3.8.

Both 120-degree structures 3.1(d) and (e) are not energetically degenerate. Indeedevaluating (3.7) with (3.8) yields

ε1 = −d3− 1− 1

2(1− d/3)(3.9)

A similar procedure for configuration 3.1(e) leads to the following

ε2 = −2d3− 1− 1

2(1 + d/3)(3.10)

Thence the energy difference

∆ε = ε1 − ε2 = − d3

1− d2 (3.11)

This implies that both configurations are quasi-degenerate at low anisotropy as theremoval of degeneracy is a third-order effect of anisotropy. Entropy can thus enablestructure with the higher energy to be stabilised at finite temperatures as observed atmean-field level.

3.2 Real-space mean-field approach

Despite the break-down of mean-field in bidimensional system, such an approach isuseful to assess symmetry analysis previously done: in this case mean-field calculationssupport a three-sublattice ordering. On a technical point of view the extension of real-space mean-field [7, 113, 19, 29] to a model with single-ion anisotropy presents someinterest as well — for example to deal with pyrochlore magnets. These technical detailscan be found in Appendix B.

Mean-field Hamiltonian reads

HMF = −∑

〈ij〉mi ·mj + d

∑

i

(Szi )2 −

∑

i

hi · Si , hi = h−∑

j(i)

mj (3.12)

where mi = 〈Si〉. Mean-field treatment yields the following system of self-consistentequations (cf Eq. B.14):

m⊥i =1

2Zi

∫ 1

−1dx√

1− x2 edx2/T eh

zix/T I1(yi) ,

mzi =1

2Zi

∫ 1

−1dx x edx

2/T ehzix/T I0(yi) , (3.13)

Zi =12

∫ 1

−1dx edx

2/T ehzix/T I0(yi) ,

3.2. REAL-SPACE MEAN-FIELD APPROACH 35

0 0.5 1temperature T

0

0.5

1

fiel

d h

P

Figure 3.2: Low-field part of mean-field phase diagram with h along the anisotropy axis andthe anisotropy strength d taken as d = 1. Spin configurations for each phase are schematicallyindicated by arrows as in Fig. 3.1. Solid and dashed lines corresponds to first- and second-order transitions, respectively.

where yi = h⊥i√

1− x2/T and In(z) is the modified Bessel function of the n-th order:

In(z) =1π

∫ π

0dα ez cosα cosnα . (3.14)

In conjunction with the self-consistent constraint

hi = h−∑

n.n.

mj (3.15)

this system of equations can be iteratively solved on L× L clusters; in this study sizesL with 3 6 L 6 12 have been used to check the stability of three-sublattice structure.Then to precisely locate phase boundaries the system (3.13,3.15) is solved assuming athree-sublattice structure.

Once convergence is achieved, various physical quantities are calculated includingthe free-energy

FMF = −J∑

〈ij〉mi ·mj − T

∑

i

lnZi , (3.16)

the internal energy EMF = 〈HMF〉, and the antiferromagnetic order parameters. Thelatter cannot be calculated as such but have to be calculated using a third-order invari-ant, which takes into account the effect of translation (Eq. (3.3)): (lα1 + ilα2 )3, wherethe real-part is zero if and only if lα1 = 0, and the imaginary part is zero if and only iflα2 = 0. By explicit calculations for clusters with 3 6 L 6 12 at all temperatures andweak magnetic fields we have verified stability of the three-sublattice structure withq0 = (4π/3, 0). After that a more detailed investigation of the H–T phase diagram hasbeen performed with the three-sublattice ansatz. Precise location of phase boundaries

36 CHAPTER 3. EASY-AXIS HTAF

in Fig. 3.2 has been determined from temperature and field scans for the antiferromag-netic order parameters indicated in Fig. 3.1 as well as for the uniform magnetization.The behaviour of the specific heat has been also used to independently verify theseresults.

At the upper transition Tc1 ≃ 1.2 in zero magnetic field only z-component of mag-netic moments become ordered. In accordance with Z6 symmetry, selection betweenvarious collinear structures is determined by the following invariant in Landau free-energy:

γ2î(lz1 + ilz2)6 + c. c.

ó. (3.17)

For negative γ2 < 0 the pure l1-state, Fig. 3.1(a), is energetically favored, while γ2 > 0corresponds to the l2-state, Fig. 3.1(b). We have verified the positive sign of γ2 in ourcase by a direct analytical expansion of Eqs. (3.13) (see App. C). Our numerical resultsalso confirm that the l2-state is stable below Tc1. Such a partially ordered phase hasa vanishing moment on one of the antiferromagnetic sublattices. A similar phase hasbeen discussed in relation to the intriguing phase diagram of Gd2Ti2O7 [111]. Here, weprovide an example, where a partially ordered phase is realized at the mean-field levelin a simple spin model.

The second transition at Tc2 ≃ 0.6 is related to the breaking of the rotationalsymmetry about z-axis. Below Tc2 the third previously disordered magnetic sublatticebecomes ordered with moments oriented within the xy plane. Simultaneously, momentsof the other two sublattices start deviating from z-axis leading to a distorted triangularstructure shown in Fig. 3.1(e). This distorted spin structure is characterized by l2 ‖ zand l1 ⊥ l2. When temperature is further decreased the coefficient γ2 in the effectiveanisotropy term changes sign at Tc3 ≃ 0.3 and one finds a first-order transition intoanother distorted triangular structure shown in Fig. 3.1(d) with l1 ‖ z.

Note, that the related model with the exchange anisotropy [84, 106] has γ2 = 0in the mean-field approximation, which leads to an additional continuous degeneracy.As a result, only two finite-temperature transitions are found in this case: from theparamagnetic state to a degenerate collinear configuration shown in Fig. 3.1(c) andthen to a degenerate distorted 120 configuration [84, 110]. Sheng and Henley [106]have discussed how different types of fluctuations, thermal, quantum, or random dilu-tion, can induce a finite γ2. For the model with the single-ion anisotropy one finds adifferent interesting possibility: the sign of the anisotropic term changes upon loweringtemperature.

The two phases in Figs. 3.1(a) and (d) have a nonvanishing total magnetisation mz.The coupling between ferro- and antiferromagnetic components is determined by theterm

mz(lz1 + ilz2)3 + c. c. , (3.18)

which is invariant under Z3 transformations (3.3). In zero magnetic field this yieldsmz ∼ (Tc−T )3/2 for states with l1 ‖ z. In contrast, states in Fig. 3.1(b) and 3.1(e) with

3.3. ZERO-FIELD BEHAVIOUR 37

l2 ‖ z have vanishing mz. This difference is important to understand the finite-fieldbehaviour, see Fig. 3.2. Magnetic field applied parallel to z-axis favours spin structureswith a finite magnetisation and stabilises states with lz1 6= 0, which is why the twointermediate low-field phases are no longer pure lz2 states. This feature is emphasisedby hatches in Fig. 3.2. The collinear-noncollinear transitions are of the second order,whereas all other transition lines are of the first order. In the case of the transitionfrom the paramagnetic state in external magnetic field the first-order nature of thetransition follows from the presence of the cubic invariant (3.18), while in other casesthe above conclusion is a consequence of the group-subgroup relation. The transitionlines intersect at a multicritical point (T ∗, h∗) = (0.6, 0.25).

The mean-field phases and the structure of the phase diagram at fields larger thanh∗ are similar to the Heisenberg triangular antiferromagnet [84] so we do not go intofurther details. As we shall see in the next section, the true thermodynamic phasesdetermined by Monte Carlo simulations of the model (3.1) differ from the mean-fieldsolutions, which is often the case in 2d. Still, the mean-field picture is expected to bequalitatively correct for 3d layered triangular antiferromagnets. By including a ferro-or antiferromagnetic interlayer coupling J ′ in the mean-field equations (3.12) and (3.13)we have verified that the predicted sequence of finite-temperature transitions remainsvalid up to |J ′/J | ∼ 0.6. For larger values of |J ′/J | we find a double transition with anintermediate l1 collinear phase similar to the previously studied case of very strong J ′

[98].

3.3 Zero-field behaviour

QLRO z LRO z QLRO

T Tc2 Tc1c3

PARA

T

Figure 3.3: Sketch of zero-field diagram as obtained by symmetry analysis and Monte Carlocalculations.

The expected sequence of finite-temperature phases is schematically shown in Fig. 3.3with three BKT-type transitions. A similar suggestion was made before for the triangu-lar antiferromagnet with the exchange anisotropy [106], though no supporting numericalresults were presented.

This scenario has been checked using Monte Carlo method on rhombic clusterswith linear size 18 6 L 6 96 for the same strength of anisotropy as in mean-fieldapproach, namely d = 1.0, for comparison (calculations at d = 0.2 which have beenused to establish phase diagram depicted in Fig. 3.8 are not presented for their being

38 CHAPTER 3. EASY-AXIS HTAF

similar). An extensive use of finite-size scaling techniques has been made to assess thevalue of critical exponents at the upper two transitions associated with the breaking ofZ6 symmetry; at the lower one, that is of BKT type, spin stiffness has been used toprecisely locate the transition. Further details on Monte Carlo technique here used canbe found in App. E and Sec. 2.4.

0.1 0.15 0.2 0.25temperature T

1

1.5

2

Bin

der

cum

ulan

t U

0z

L = 18L = 24L = 36L = 48L = 72

0.35 0.4 0.45temperature T

1

1.05

1.1

Bin

der

cum

ulan

t U

qz

L = 18L = 24L = 36L = 48L = 60L = 96

Figure 3.4: Results at h = 0 for the easy-axis HTAF with d = 1.0. Temperature dependenceof Binder’s cumulant associated with uniform magnetisation mz for different cluster sizes(left): the crossing indicates Tc2. Temperature dependence of Binder’s cumulant associatedwith antiferromagnetic order parameter mzq for different cluster sizes (right): the mergingindicates Tc1.

To exploit data obtained during Monte Carlo calculations various observables andtechniques can be used depending on the specific sought features as explained in Section2.4. In this case as the first symmetry breaking is expected along z, observables alongz are used: Binder’s cumulant only yields a rough estimate of transition temperaturesas shows Fig. 3.4: Tc1 ≈ 0.4 and Tc2 ≈ 0.2. For these transitions to be more accuratelylocated the scaling of spin susceptibility is finer. First the value of critical exponent η ischecked using a procedure that doesn’t require a precise location of the transition [70, 8]:it consists in the plotting of scaled susceptibility χzq/L

2−η versus Binder’s cumulantU zq . In Fig. 3.5 the exponent η yielding the best fits is shown. The obtained valuesη1 = 0.26± 0.01 and η2 = 0.12± 0.01 are in very good agreement with renormalisationgroup predictions: η1 = 1/4 and η2 = 4/62 = 1/9 [40]. Next, scaled susceptibility isused to precisely locate transition temperatures as illustrated in Fig. 3.6.

Last, the third transition is associated with the breaking of S1 and thence expectedas BKT-like. To deal with a BKT transition spin stiffness is very well adapted asalready discussed in Section 2.4. Furthermore the very fact that it exhibits a jumpthe height of which is a universal value, namely 2TBKT/π for integer vortices, at thetransition clearly identify the transition as BKT. Using this method with the simplestform of scaling to extract Tc3 as illustrated in Fig. 3.7 the transition temperature is

3.4. FINITE-FIELD BEHAVIOUR 39

1.01 1.03 1.05Binder cumulant U

q

z0.2

0.4

scal

ed s

usce

ptib

ility

χqz

/L2-

ηL = 18L = 24L = 36L = 48L = 96

η = 0.26

1.002 1.004 1.006

Binder cumulant Uq

z

1

1.5

2

2.5

3

scal

ed s

usce

ptib

ility

χqz

/L2-

η L = 18L = 24L = 36L = 60L = 72

η = 0.12

Figure 3.5: Results at h = 0 for the easy-axis HTAF with d = 1.0. Scaled susceptibilityplotted versus Binder cumulant in the vicinity of Tc1 (left) and Tc2 (right). The indicatedvalue of critical exponent η is the one that realises the best collapse of all data points onto asingle curve.

Tc3 = 0.168± 0.001.

3.4 Finite-field behaviour

At finite field the symmetry of easy-axis HTAF Hamiltonian is the same as the sym-metry of HTAF Hamiltonian: phase diagram should consequently look similar with thesame stabilised phases and the same nature of transitions. Yet due to the anisotropysome differences are expected. As easy-axis anisotropy reduces transverse fluctuationsthe domain of ordered phases should stretch to higher temperatures. Concerning thevery-low field region as zero-field behaviour in both systems is totally different the waystransition lines close at h = 0 are different as well: it can be expected in the case ofeasy-axis HTAF that lower BKT transition between up-up-down collinear structure anddistorted 120-degree structure closes at the finite-temperature zero-field BKT transi-tion that exists in this system. Similarly the upper transition between collinear andparamagnetic phases that belongs to three-state clock-model universality class closesat zero-field upper transition, that is associated whith Z6 symmetry: group-subgrouprelation enables such a junction of transitions.

Another effect of anisotropy is to enlarge one-third magnetisation plateau at zerotemperature. This appears as a splitting of the single transition existing in HTAF; thesplitting can be estimated by linearising zero-temperature energy about equilibriumconfigurations in terms of small deviations. Let’s start with the transition betweendistorted 120-degree structure and collinear one. Eq. (3.6) yields for up-up-down

40 CHAPTER 3. EASY-AXIS HTAF

0.34 0.36 0.38 0.4temperature T

0.32

0.36

0.4sc

aled

spi

n su

scep

tibili

ty χ

qz /L2-

η

L = 18L = 24L = 36L = 60

0.18 0.19 0.2 0.21temperature T

0.32

0.36

0.4

scal

ed s

usce

ptib

ility

χqz /L

2 -η

L = 24L = 36L = 48L = 60L = 72L = 96

Figure 3.6: Results at h = 0 for the easy-axis HTAF with d = 1.0. Scaled susceptibilityas a function of temperature; scaling with η = 0.26 at the upper transition (left) and withη = 0.12 at the second transition (right).

structure:

εuud = −1− d− h3

(3.19)

For a distorted 120-degree structure as depicted in Fig. 2.1 Eq. (3.6) yields

ε(θ) = cos 2θ − 2 cos θ − d3

Ä1 + 2 cos2 θ

ä− h

3(−1 + 2 cos θ) (3.20)

Minimisation according to θ yields either θ = 0 mod π, which is up-up-down collinearstructure, or

cos θ0 =1 + h/3

2(1− d/3)(3.21)

and the associated energy

ε(θ0) = − (1 + h/3)2

2(1− d/3)+

2d3

+ εuud (3.22)

A straightforward analysis shows that the first part of right-hand-side term is negativefor 0 6 d 6 dc = 3/2, where dc is the upper limit of stability for a distorted 120-degree structure to exist; consequently the limit of stability is the one imposed by theexpression of cos θ0:

hc1 = 3− 2d (3.23)

For d = 0.2, which has been used for Monte Carlo simulations, hc1 = 2.6.As for the upper bound of the splitting about h = 3 it is obtained linearising the

energy of V-shape configuration (see Fig. 2.2) about (α, β) = (π, 0).

εV (α, β) = 1+2 cosα cos β−2 sinα sin β−d3

(2 cos2 β+cos2 α)−h3

(2 cosβ+cosα) (3.24)

3.4. FINITE-FIELD BEHAVIOUR 41

0 0.1 0.2temperature T

0

0.1

0.2

0.3

spin

stif

fnes

s ρ

s

L = 24L = 36L = 60L = 96

0 0.02 0.041/L

0.165

0.17

T

Figure 3.7: Results at h = 0 for the easy-axis HTAF with d = 1.0. Spin stiffness for differentcluster sizes. Tc3 is determined by the scaling of T (L), where T (L) such that ρS(T (L), L) =2T (L)/π – these are the temperatures of the crossing points between the solid line and curvesrepresenting ρS . The inset depicts the extrapolation made to 1/L = 0.

Introducing γ = π − α, the energy can be expanded as

εV = εuud + (β, γ)Ç

1 + 2d3

+ h3

−1−1 1 + d

3− h

6

åÇβγ

å(3.25)

This yields

hc2 =3 +

»9 + 8(2d2 + 9d)

2(3.26)

For d = 0.2, hc2 = 3.95.Zero-field saturation field is also reduced due to easy-axis anisotropy. The saturation

field can be determined developing εV around (α, β) = (0, 0):

εV = εsat + (α, β)Ç −1 + d

3+ h

6−1

−1 −1 + 2d3− h

3

åÇαβ

å(3.27)

where εsat = 3 − d − h is the energy of a fully polarised configuration. Hence thesaturation field:

hsat = 9− 2d (3.28)

For d = 0.2 this is hsat = 8.6Numerical determination of transitions is carried out with a similar Monte Carlo

algorithm as previously described; the observables I used are the same as in the case

42 CHAPTER 3. EASY-AXIS HTAF

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5temperature T

0

1

2

3

4

5

6

7

8

9fi

eld

hA

C

B

B’

Tc1

Tc2

Tc3

Figure 3.8: Phase diagram of easy-axis HTAF calculated for d = 0.2. Solid lines are a guideto the eye linking calculated points. Spin structures are sketched by the configuration on atriangular plaquette with z axis assumed vertical and xy plane perpendicular to the plane ofthe sheet.

of HTAF as the transitions belong to the same classes of universality. Let’s review thedifferent transitions and illustrate for each of them their determination. At low fieldupper transition CTc1 is investigated monitoring z spin component susceptibility χzq ,scaled as χzq/L

2−η with η = 4/15 that is the critical exponent for the class of universalityof three-state clock model, and Binder’s cumulant U zq .

The lower transition B′Tc3 corresponds to the ordering of transverse component: itis thus located using transverse spin component susceptibility χ⊥q , scaled as χ⊥q /L

2−η

with Fisher’s exponent equal to 1/4 that is the value for a BKT transition; Binder’scumulant U⊥q is used for cross-checking.

Transition BC between up-up-down phase and high-field phase can be located usingtransverse spin component susceptibility χ⊥q scaled as χ⊥q /L

2−η with η = 1/4 that is thevalue for a BKT transition, and cross-checking with Binder’s cumulant U⊥q . Given the

3.4. FINITE-FIELD BEHAVIOUR 43

3.9 4 4.1 4.2 4.3 4.4 4.5field h

0

0.2

0.4

0.6

0.8

1

scal

ed s

pin

susc

eptib

ility

χqt /

L2-

η

L = 24L = 36L = 48L = 60

η = 1/4

Figure 3.9: This plot depicts scaled spin susceptibility along transverse component χ⊥q /L2−η

in a field scan at T = 0.27. Two transitions can clearly be seen: this denotes the roundig ofthe transition line backward to multicritical C point.

quasi horizontal shape of the transition it is more adapted to sweep field at constanttemperature rather than the opposite; Fig. 3.9 illustrates such a sweep in the vicinityof C point, which clearly shows the double transition that reflects the rounded shapeof the transition line in this vicinity.

Double transition associated with the breaking of S1 ⊗ Z3 between paramagneticor saturated phase presents the same hardship to be located as is the case in HTAFdue to the breaking along a single spin component. As in that case, phase transitionassociated with the breaking of S1 is located thanks to finite-size scaling of spin stiffnesswhereas the transition associated with discrete symmetry breaking is located thanks tothe scaling of transverse spin component susceptibility χ⊥q /L

2−η with η = 1/4. Binder’scumulant can be used to further confirm the location of the upper transition: thelower one cannot be seen with this observable as the range of its value for differentcluster sizes is dramatically reduced so that plots for different cluster sizes appear ontop of each other. Such a procedure is illustrated in Fig. 3.10. As can readily beobserved comparing Figs. 3.8 and 2.3 the separation between two transitions BC islarger in case of the easy-axis HTAF than in case of HTAF, which is a consequence ofthe reduced linear energy of domain walls in the anisotropic system where transversespin fluctuations are hindered.

Another point of interest is to question the evolution of zero-field behaviour at tinyfield. As previously seen an intermediate critical phase exists for Tc2 6 T 6 Tc1: thisphase probably extends to finite fields for very weak fields scaling with d3 which is theorder of degeneracy between configurations with and without phase locking (angle ϕ inEq. 3.5). This is however a field range out of reach for this numerical investigations.

44 CHAPTER 3. EASY-AXIS HTAF

0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1temperature T

1.8

2

2.2

2.4

2.6

2.8

scal

ed s

pin

susc

eptib

ility

χqt /

L2-

η L = 24L = 36L = 48L = 60L = 72

η = 4/15

0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1temperature T

1.005

1.01

1.015

1.02

Bin

der

cum

ulan

t U

qt

L = 24L = 36L = 48L = 60L = 72L = 96

0.1 0.105 0.11 0.115 0.12 0.125temperature T

0.04

0.06

0.08

0.1

0.12

0.14

spin

stif

fnes

s ρ

S

L = 24L = 36L = 48L = 60L = 72L = 96

20 40 60 80 100linear size L

0.1

0.105

0.11

0.115

0.12

0.125

0.13

tem

pera

ture

TL

Figure 3.10: Location of successive transitions at h = 7.0. Upper transition associatedwith the breaking of Z3 is located thanks to the scaling of transverse component of spinsusceptibility χ⊥q /L

2−η with η = 4/15 (upper left); cross-checking with Binder’s cumulant

U⊥q (upper right). Lower transition associated with the breaking of S1 is located using spinstiffness: temperature TL such that ρS(TL, L) = (2/π)TL (lower left) is scaled according toEq. D.21 (lower right): the obtained parameters for this nonlinear fit are TBKT = 0.092 andc = −1.25.

Chapter 4

Heisenberg TriangularAntiferromagnet with easy-planesingle-ion anisotropy

Determining the phase diagram of Heisenberg triangular antiferromagnet with easy-plane single-ion anisotropy necessarily induces an interest in the phase diagram of XYtriangular antiferromagnet: the former shares the same symmetries as the latter whichis the paradigmatic model of this class of systems. Even though XY triangular antifer-romagnet has been studied a lot since the 1980’s either directly [85, 66, 46, 26, 57, 60,56, 67, 68, 134, 69, 91, 90] or indirectly by the study of closely related bidimensionalmodels such as fully frustrated XY model, associated with Josephson junction arraysat half flux per plaquette [107, 108, 37, 59], and XX0 model [16, 15, 90], a detaileddiscussion of its whole phase diagram is still welcome as it clarifies many questions leftso far unanswered as far as I know.

In this chapter the phase diagram of XY triangular antiferromagnet (XY TAF) isthus discussed in some details (see Fig. 4.1) before dealing with Heisenberg triangularantiferromagnet with easy-plane single-ion anisotropy (easy-plane HTAF). First zero-field behaviour is described (Sec. 4.1); then the discussion of finite-field phase diagram issplit into two parts: the first one deals with moderate fields and the delicate question ofthe connection of finite-field transition lines with zero-field transition points (Sec. 4.2);the second one presents high-field behaviour, which means between one-third magneti-sation field and saturation field (Sec. 4.3). Last the diagram of Heisenberg triangularantiferromagnet with easy-plane single-ion anisotropy is presented (Sec. 4.4).

45

46 CHAPTER 4. EASY-PLANE HTAF

0 0.1 0.2 0.3 0.4 0.5 0.6temperature T

0

1

2

3

4

5

6

7

8

9

fiel

d h

0.5 0.505 0.51 0.515

0

0.2

0.4

0.6

0.8

D

D

C

A

B

TBKT

Tc

Figure 4.1: Phase diagram ofXY TAF. Lines are guides to the eye between calculated points.The inset offers an enlarged view about low-field multicritical point D. Spin structures aresketched by the configuration on a triangular plaquette with x axis assumed horizontal and yaxis vertical in the plane of the sheet. Transition line AC is split into two with an intermediatecritical phase.

4.1 XY triangular antiferromagnet at zero-field

Even though a continuous symmetry cannot be broken in bidimensional systems asdemonstrated by Mermin and Wagner [77], another kind of phase transition is possible:it is induced by stable topological excitations as found out by Berezinskii on the onehand [10, 11], and by Kosterlitz and Thouless on the other hand [61] – which is referredto as a BKT transition. Interestingly some bidimensional systems intertwine bothcontinuous and discrete symmetries, which creates highly non-trivial phase transitionscenarios. This is the case of classical XY TAF at zero field:

H =∑

〈ij〉Si · Sj (4.1)

4.1. XY TRIANGULAR ANTIFERROMAGNET AT ZERO-FIELD 47

where the Hamiltonian has been normalised by the coupling exchange and by spin aswell so that Si in the above expression is a classical bidimensional unit vector. Atlow-temperature this Hamiltonian admits a 120-degree structure that can be describedas:

〈Si〉 = l1 cos (q · ri) + l2 sin (q · ri) (4.2)

with (l1, l2) a pair of orthogonal vectors and q = (4π/3, 0) the three-sublattice orderingwave vector. Beside continuous S1 symmetry associated with the global rotation ofspins, that is the rotation of the pair (l1, l2), a discrete Z2 symmetry comes into thegame, which corresponds to the direct or indirect character of the basis (l1, l2). Inreal space it corresponds to the fact spins in this non-collinear structure can rotateeither clockwise or anticlockwise around a triangle. This extra discrete symmetry,called chirality, was pointed out in 1984’s pioneering works [85, 66]. Due to the discretecharacter of Z2, its breaking does induce a long-range order. The subsequent questionsare then: what is the nature of the ordering? Does chiral order come with quasi-long range magnetic order? Does a magnetically disordered chiral phase exist? Ifan early agreement was reached on the existence of a quasi-long range magneticallyordered phase with long-range chiral order, the scenario for ordering has remained acontroversial issue for more than twenty years with proponents of two transitions [85,68, 134, 69, 91] opposing those in favour of a single one belonging to a new universalityclass [66, 67]. This latter proposition was made possible by the hardship to numericallyresolve both transitions as they are quite close by – the latest estimate [95] publishedbefore this work [74] yields Tchiral = 0.512(1) and TBKT = 0.508(1), which agrees bothwith certain older results and with our own ones, and participates to today’s consensuson the number of transitions. Renormalisation group approaches for frustrated magnetsare at best difficult, at worst misleading, especially in the vicinity of d = 2, which isthe one of interest here. While a general agreement on the BKT nature of the lowertransition exists, the nature of the upper transition is regularly debated [85, 66, 69, 93,91]. The point is that critical exponents are notoriously hard to compute. Symmetryarguments make Ising universality class the most natural, which was pointed up fromthe very beginning [85]. Yet numerical investigations have failed so far to calculatecritical exponents matching those of Ising universality class (see Table II in [95] for asummary of results). As often noted misleading finite-size effects could be responsiblefor these difficulties [91, 112]. In the following, specific heat is studied as the logarithmicscaling of its peak value with cluster linear size L is a clear argument in favour of Isinguniversality class.

To investigate chiral ordering the most adapted parameter is chiral susceptibility:

χκ = 〈κ2〉 − 〈κ〉2 (4.3)

where chirality κ reads

κ =1L2√

3

[

∑

∆

ε∆ (S1 × S2 + S2 × S3 + S3 × S1)

]

· z (4.4)

48 CHAPTER 4. EASY-PLANE HTAF

with summation running over triangles, spins numbered clockwise around a triangle,and ε∆ = ±1 taking into account staggered ordering. In this definition κ is really ascalar: the cross product are all along z as spins are strictly within xy plane. With thisdefinition κ is a non-zero Ising variable in phases with chiral order: introducing spinstructure (4.2) in (4.4) yields (there are 2L2 triangles)

κ = (l1 × l2) · z (4.5)

which is zero in phases without chiral order (l1 and l2 collinear or one of the two zero)and is ±1 in 120-degree phase, even if distorted (distortion doesn’t change l1 and l2

but m). Chiral susceptibility can be scaled as χκ/L2−η, which quantity plotted againsttemperature (see App. D) yields the transition temperature. Of course this requiresη to be known, which is a matter of concern especially in this case as the nature ofchiral transition is much debated. As explained in App. D it is possible to determinate ηplotting χκ/L2−η vs Uκ adjusting η so that data points for different cluster sizes collapseonto a single curve. Yet in this case due to the closeness of transitions it appears thismethod is not satisfactory with cluster sizes here used. As critical exponents differ verymoderately between several class of universality in 2d, sorting out this question canindeed be quite tricky. The argument here put forward makes use of a specificity of Isinguniversality class that does dramatically distinguish it from other universality classes:α = 0, which implies specific heat is logarithmically divergent. Obtained results tend toexclude α > 0 as can be seen in log-log plot presented in Fig. 4.2: the clear downwardbending cannot be compatible with α > 0. My viewpoint is that numerical estimationsof other critical exponents are not yet satisfactory because too small systems are used.Large clusters would be welcome not to yield high precision estimates as is the casein most problems but right ones: as discussed in [91] and supported by Fig. 6 in thatarticle, too small clusters most probably exhibit a misleading behaviour and an ideaabout thermodynamic limit can be gained only using very large clusters. Consequentlythe assumption η = 1/4 according to Ising universality class value is used due totheoretical arguments above mentioned, in agreement with symmetry considerations,and numerical support gained from the analysis of specific heat. Fig. 4.3 shows thisscaling and the associated temperature: Tchiral = 0.512± 0.001.

As far as magnetic ordering temperature is concerned usual means of locating aBKT transition can be used: spin stiffness, spin susceptibility, and Binder’s cumulant.Spin susceptibility χq, which is the total spin susceptibility

χq =1N2s

∑

i,j

〈Sxi Sxj + Syi Syj 〉

Teiq·(ri−rj) (4.6)

is used as illustrated in Fig. 4.3.

4.2. LOW FIELDS: COMPETING Z2 AND Z3 SYMMETRY BREAKING 49

100linear size L

10

spec

ific

hea

t max

imum

cm

ax

Figure 4.2: Maximum of the specific heat at h = 0 plotted vs cluster linear size L in alog-log plot: if critical exponent α were positive data points should align on a straight line.As can be observed a clear downward bending exists that prevents α > 0.

4.2 Low fields: competing Z2 and Z3 symmetry break-ing

Let’s now apply a magnetic field along x direction:

H =∑

〈i,j〉Si · Sj − h

∑

i

Sxi (4.7)

At finite field XY triangular antiferromagnet retains a continuous degeneracy. In-deed Hamiltonian (4.7) can be written taking into account a three-sublattice patternas:

H =12

∑

∆

ÇS1 + S2 + S3 −

hx3

å2

+ const. (4.8)

This expression clearly shows that ground-states are characterised by the following setof constraints:

S1 + S2 + S3 =hx3

(4.9)

From this relation the value of saturation field is immediately obtained as h = 9. Asthere are three degrees of liberty (three bidimensional unit vectors) and two constraints,one free parameter remains: a continuous degeneracy persists. As this degeneracy is notrelated to a symmetry in the Hamiltonian (4.7), it can be expected thermal fluctuations

50 CHAPTER 4. EASY-PLANE HTAF

0.502 0.504 0.506 0.508 0.51temperature T

0.46

0.48

0.5

0.52

0.54

0.56

scal

ed s

pin

susc

eptib

ility

χq /L

2-η L = 24

L = 48L = 72L = 96L = 120L = 150

0.509 0.51 0.511 0.512 0.513 0.514 0.515temperature T

2

3

4

5

6

7

scal

ed c

hira

l sus

cept

ibili

ty χ

κ /L2-

η

L = 24L = 48L = 72L = 96L = 120L = 150L = 180

Figure 4.3: The upper transition temperature is located thanks to chiral susceptibility(right); the lower one with spin susceptibility (left).

lift it. This is indeed the case [46, 57, 102]: symmetry is reduced to the compositediscrete Z2 ⊗ Z3 symmetry.

As mentioned in the introduction to this chapter the discussion first focuses onlow-field part of the diagram, namely for h < 3. At h = 3 one-third magnetisationplateau appears in magnetisation curves: it corresponds to an up-up-down structureas can be seen from (4.9). This collinear structure is stable at T = 0 and h = 3as shown in [46]. At finite temperature thermal fluctuations do widen the field rangeof stability of up-up-down structure mainly towards h < 3 as increased fluctuationsof spin component transverse to applied field destabilise 120-degree structure. Hencea twofold symmetry-breaking scenario: first Z3 is broken with the collinear ordering;then Z2 symmetry with the ordering into 120-degree structure. This scenario impliesthe successive symmetry-breakings occur along different components: Z3 along fielddirection x and Z2 in transverse direction – or according to chiral order parameter: inthis case both magnetic ordering along y and chiral ordering happen simultaneouslyas 120-degree structure has an inherent chiral symmetry. Such a separation makes iteasier to numerically assess the scenario as there is a clear distinction among orderparameters, which may otherwise be problematic in case of multiple ordering along asingle component. According to the exact results of the hard hexagon model [9] theupper transition belongs to three-state clock model universality class.

Let’s now detail how this numerical investigation has been done. For the uppertransition χxq can be scaled as χxq/L

2−η using η = 4/15, the value of the universalityclass of three-state clock model where

χαq =〈ÄSαqä2〉 − 〈Sαq 〉2T

withα ∈ x, y (4.10)

This assumption theoretically grounded in Baxter’s results, can also be numericallysupported in various ways using Binder’s cumulant or the ratio of correlation functions

4.2. LOW FIELDS: COMPETING Z2 AND Z3 SYMMETRY BREAKING 51

0.48 0.5 0.52 0.54 0.56temperature T

0

0.2

0.4

0.6

0.8

1

scal

ed s

usce

ptib

ility

χqx /

L2-

ηL = 48L = 60L = 84

η = 4/15

0.5 0.6 0.7 0.8 0.9 1

correlation function ratio Γx

0

0.2

0.4

0.6

0.8

1

scal

ed s

pin

susc

eptib

ility

χqx

/ L2-

η L = 48L = 60L = 84

η = 4/15

Figure 4.4: Upper transition location at h = 1.0 Scaled spin susceptibility χxq/L−(2−η) with

η = 4/15 is used to locate the upper transition (left). The correctness of the value of η ischecked plotting scaled spin susceptibility χxq/L

−(2−η) vs the ratio of correlation function Γx.The value of three-state clock-model universality class is fully satisfactory to have data pointscollapse onto a single curve (right).

Γx (see App. A, Eq. (2.19), and the discussion on the properties of this ratio): eitherof them can be used to locate transition temperature as well, and even though Binder’scumulant may be less accurate the estimate thus obtained should be reasonably closeto the one resulting from the scaling of susceptibility; second to calculate η as explainedin App. D. Such a mix of techniques is illustrated in Figs. 4.4 and 4.5 at h = 1.0.

The lower transition can be investigated in two different ways: according to chiralityor to magnetic ordering along transverse direction. Both yield the same transition, instrong support for the explicited scenario that implies simultaneous chiral and trans-verse magnetic ordering at the lower transition as shows Fig. 4.6. Techniques are thesame as previously explained at zero field for chirality; concerning transverse magneticordering χyq is scaled using η = 1/4 (Ising universality class).

A delicate problem arises when drawing the phase diagram down to zero field: zero-field symmetry-breaking pattern is different from the one above described at moderatefinite field. How can both be connected? Finite-field upper transition is a Z3 symmetry-breaking line: it cannot join zero-field upper transition which is a Z2 symmetry-breakingpoint. On the contrary finite-field lower transition that corresponds to chiral orderingcan join zero-field upper transition: this is even the most natural scenario as it is thesimplest way to divide (T, h) plane into a chirally ordered part and a disordered one.This scheme implies the existence of a multicritical point D defined by (TD, hD) withhD 6= 0. At D an inversion between Z3 and Z2 symmetry-breakings takes plase: forh > hD the former occurs first whereas for h < hD the latter does, which impliesthat a magnetically disordered chiral phase is expected at very low field. Numericalevidences of this scheme are sought using chiral susceptibility χκ on the one hand andspin susceptibility along x χxq on the other hand. Binder’s cumulant and y component

52 CHAPTER 4. EASY-PLANE HTAF

0.46 0.48 0.5 0.52 0.54 0.56temperature T

1

1.2

1.4

1.6

1.8

2

Bin

der

cum

ulan

t U

qx

L = 48L = 60L = 84

0.46 0.48 0.5 0.52 0.54temperature T

0.6

0.7

0.8

0.9

1

corr

elat

ion

func

tion

ratio

Γx

L = 48L = 60L = 84

Figure 4.5: Upper transition location at h = 1.0. Binder’s cumulant Uxq (left) is comparedwith the ratio of correlation functions Γx (right). In this case both exhibit similar precisionand yield following transition temperature: 0.520± 0.005.

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54temperature T

0

0.2

0.4

0.6

0.8

1

scal

ed s

usce

ptib

ility

χκ /

L2-

η

L = 48L = 60L = 84

η = 1/4

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54temperature T

0.2

0.4

0.6

0.8

1ra

tio o

f co

rrel

atio

n fu

nctio

ns Γ

ψ L = 48L = 60L = 84

Figure 4.6: Lower transition at h = 1.0. Finite-size scaling of chiral susceptibility (4.3)(left) to locate Ising transition into chiral ordered phase and ratio of correlation functions Γy

(right) to illustrate simultaneous ordering along transverse direction.

quantities are monitored as well. For illustration results at h = 0.1 are proposed inFig. 4.7. Measurements at h = 0.75 and h = 0.5 assures 0.5 6 hD 6 0.75. At h = 0.75the location of transitions carried out in a similar way to the case h = 0.1 yieldsTchiral = 0.503 ± 0.001 and TZ3

= 0.507 ± 0.001 (see Fig. 4.8) whereas at h = 0.5 theorder is reversed: Tchiral = 0.509± 0.001 and TZ3

= 0.506± 0.001 (see Fig. 4.9). In thelatter case chiral transition can be located only with chiral susceptibility or observablesbuilt upon susceptibility: magnetic ordering is indeed disconnected from chiral orderingin this regime as can readily be seen noticing there is no difference between x andy component in their ordering temperature (see Fig. 4.10). As a consequence anintermediate magnetically disordered chiral phase exists in this regime.

Last point not yet discussed about this low-field region is the way the lower transition

4.3. BREAKING OF Z6 SYMMETRY AT HIGH FIELDS 53

0.5 0.502 0.504 0.506 0.508 0.51 0.512 0.514 0.516 0.518 0.52temperature T

0.2

0.4

0.6

0.8

1

scal

ed s

usce

ptib

ility

χκ /

L2-

ηL = 48L = 60L = 84L = 168

η = 1/4

0.5 0.502 0.504 0.506 0.508 0.51 0.512 0.514 0.516temperature T

0.4

0.45

0.5

0.55

scal

ed s

usce

ptib

ility

χqx /

L2-

η

L = 48L = 60L = 84L = 168

η = 4/15

Figure 4.7: Finite-size scaling of chiral susceptibility (4.3) (left) to locate Ising transitioninto chiral ordered phase and finite-size scaling of χxq (right) to locate the transition associatedwith the breaking of Z3 symmetry; both at h = 0.1.

terminates at zero field. Actually it is fully admissible that it terminates at zero-fieldBKT transition as Z3 is a subgroup of S1. As low-field spin-wave expansion of thefree-energy yields a three-fold-anisotropy-like term [46, 74], which is irrelevant aboutBKT transition in [40], finite-field transition line necessarily closes at zero-field BKTtransition.

4.3 Breaking of Z6 symmetry at high fields

For h > 3 the behaviour is quite different. Indeed up-up-down structure is no longerstable as constraint (4.9) clearly shows: in this equation there are indeed two regimesdepending on whether h 6 3 or h > 3. In the latter regime stable structures exhibitboth the breaking of sublattice symmetry (which means the breaking of Z3) and ofthe symmetry along transverse direction (therefore breaking of Z2) [46, 57]. Thesestructures can be described as

〈Si〉 = l1 cos (q · ri) + m (4.11)

where m is uniform magnetisation. Only y component is critical. This has a strikingconsequence on the breaking of composite Z2 ⊗ Z3 discrete symmetry: it is broken asa whole Z6 symmetry. If Korshunov already proposed such a possibility for this high-field transition [57], the proposition hadn’t found any further support, either theoreticalor numerical, before the results here presented (see also [74]). The hallmark of suchsymmetry-breaking as compared to other possible symmetry-breakings in this systemis the existence of an intermediate critical phase with algebraically decaying correla-tion functions between disordered and ordered phase [40]. As a consequence the mostconvincing evidence for this scenario is the existence of the intermediate critical phase.

54 CHAPTER 4. EASY-PLANE HTAF

0.48 0.485 0.49 0.495 0.5 0.505 0.51 0.515 0.52temperature T

0.2

0.4

0.6

0.8

1

scal

ed s

usce

ptib

ility

χκ /

L2-

ηL = 48L = 84L = 132

η = 1/4

0.49 0.495 0.5 0.505 0.51 0.515 0.52temperature T

0.2

0.3

0.4

0.5

0.6

0.7

0.8

scal

ed s

pin

susc

eptib

ility

χqx /

L2-

η L = 48L = 84L = 132

η = 4/15

Figure 4.8: Location of Z2 and Z3 symmetry breakings at h = 0.75. The former is associatedwith chiral ordering and is thus located using chiral susceptibility χκ scaled as χκ/L

2−η

with η = 1/4 (Ising universality class). The latter corresponds to the breaking of latticetranslational symmetry along the applied field and is thus located using spin susceptibilityscaled as χxq/L

2−η with η = 4/15 (three-state clock model universality class). These resultsshow hD 6 0.75.

Such an evidence can be obtained from the use of the ratio of correlation functions intro-duced in Eq. (2.19) Γy, which has the same properties as Binder’s cumulant but lowernoise as it is a second-order quantity (see App. D). In case of Z6 symmetry-breaking asis here investigated, this ratio shows both upper and lower transitions which is not thecase with Binder’s cumulant. This is neatly illustrated in Fig. 4.12 at h = 5.0 wherethe width of the intermediate phase is rather large. For transition temperatures to bemore accurately calculated the scaling of spin susceptibility χyq can then be used withη = 1/4 at the upper transition and η = 4/62 = 1/9 at the lower one [40]. Figs. 4.11and 4.12 shows the results for h=5.0.

Another transition remains for the discussion: the one between high-field phaseand collinear phase. From the latter to the former a Z2 symmetry-breaking occurs.As shown by Kawamura there is no spin-flop transition but a continuous one. Hencethe transition is expected to be of Ising type. Results obtained through field scansacross the transition for transverse spin susceptibility and Binder’s cumulant supportthis assumption.

In the proposed phase diagram another multicritical point, called C in Fig. 4.1,appears. I haven’t investigated it in details. What can be said is that transition lineCD and upper transition line AC merge at C with different slopes as they correspondto different representations. Lower transition line AC is also expected to merge atC: indeed the splitting of the transition with the existence of an intermediate criticalphase is observed in all the measurements; the occurrence of a first-order transitionin the small field range 3.4 6 h 6 3.8 seems improbable. Last Ising line BC ends atmulticritical point C as well.

4.3. BREAKING OF Z6 SYMMETRY AT HIGH FIELDS 55

0.5 0.505 0.51 0.515 0.52temperature T

0

0.2

0.4

0.6

0.8

1

scal

ed s

usce

ptib

ility

χκ /

L2-

η

L = 48L = 84L = 132

η = 1/4

0.49 0.495 0.5 0.505 0.51 0.515 0.52temperature T

0.3

0.4

0.5

0.6

0.7

scal

ed s

pin

susc

eptib

ility

χqx

/ L2-

η L = 48L = 84L = 132

η = 4/15

Figure 4.9: Location of Z2 and Z3 symmetry breakings at h = 0.5. The former is associatedwith chiral ordering and is thus located using chiral susceptibility χκ scaled as χκ/L

2−η

with η = 1/4 (Ising universality class). The latter corresponds to the breaking of latticetranslational symmetry along the applied field and is thus located using spin susceptibilityscaled as χxq/L

2−η with η = 4/15 (three-state clock model universality class). These resultsshow hD > 0.5.

0.495 0.5 0.505 0.51 0.515temperature T

0.2

0.3

0.4

0.5

scal

ed s

pin

susc

eptib

ility

χqy

/ L2-

η L = 48L = 84L = 132

η = 4/15

Figure 4.10: Scaling of spin susceptibility along y direction χyq as χyq/L2−η with η = 4/15

which corresponds to the breaking of lattice translational invariance associated with magneticordering. Comparison with scaled susceptibility along x direction χxq/L

2−η with η = 4/15in Fig. 4.9 shows that for h 6 hD the lower transition corresponds to magnetic orderingalong both component, which is an evidence of the existence of an intermediate magneticallydisordered chiral phase in this regime.

56 CHAPTER 4. EASY-PLANE HTAF

0.462 0.464 0.466 0.468 0.47 0.472temperature T

0.56

0.58

0.6

0.62

0.64

0.66

scal

ed s

usce

ptib

ility

χqy /

L2-

η

L = 48L = 60L = 84L = 120

η = 1/4

0.391 0.392 0.393 0.394 0.395temperature T

0.626

0.628

0.63

0.632

0.634

0.636

0.638

0.64

scal

ed s

usce

ptib

ility

χqy /

L2-

η

L = 48L = 60L = 84L = 120

η = 1/9

Figure 4.11: Scaled spin susceptibility χyqL−(2−η) with η = 1/4 (upper transition; left

panel) and η = 1/9 (lower transition; right panel) to locate double transition of Z6 symmetrybreaking at h=5.0.

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5temperature T

0.85

0.9

0.95

1

corr

elat

ion

func

tions

rat

io Γ

y

L = 48L = 60L = 84L = 168

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5temperature T

1

1.02

1.04

1.06

1.08

1.1

Bin

der

cum

ulan

t U

qy

L = 48L = 60L = 84L = 168

Figure 4.12: Correlation ratio (2.19) for critical y component at h = 5.0 (left). The in-termediate phase with algebraically decaying correlations corresponds to the intermediatetemperature range over which plots for different cluster sizes L merge. Long-range orderedphase with exponentially decaying correlation can also be identified as the lowest temper-ature range with separated plots; the highest one corresponds to paramagnetic phase. Forcomparison Binder’s cumulant is shown as well (right).

4.4. EASY-PLANE HTAF 57

4.4 Heisenberg triangular antiferromagnet with easy-plane single-ion anisotropy

0 0.1 0.2 0.3 0.4 0.5temperature T

0

1

2

3

4

5

6

7

8

9

fiel

d h

0.355 0.36 0.365 0.37

0

0.1

0.2

0.3

0.4

B C

D

A

TBKT

Tc

D

Figure 4.13: Phase diagram of easy-plane HTAF calculated at d = −0.2. Lines are guidesto the eye between calculated points. The inset offers an enlarged view about low-field multi-critical point D. Spin structures are sketched by the configuration on a triangular plaquettewith x axis assumed horizontal and y axis vertical in the plane of the sheet. Transition lineAC is split into two with an intermediate critical phase.

The Hamiltonian of Heisenberg triangular antiferromagnet with easy-plane single-ion anisotropy reads

H =∑

〈i,j〉Si · Sj − d

∑

i

(Szi )2 − h

∑

i

Sxi (4.12)

with d < 0. It belongs to the same universality class as XY triangular antiferromagnet:same symmetries are involved. Given the extra fluctuations in z direction the whole

58 CHAPTER 4. EASY-PLANE HTAF

phase diagram should shrink to lower temperatures. Phase transitions have been locatedas in XY triangular antiferromagnet: details of this location are therefore not repeatedhereafter, which would not bring anything new.The phase diagram has been calculatedat d = −0.2: it is depicted in Fig. 4.13.

The splitting of zero-field transitions exhibits little dependence on the strength d ofthe anisotropy as shows the comparison between d = −0.2 and d = −0.4: for d = −0.2,Tchiral = 0.366±0.0005 and TBKT = 0.362±0.0005; for d = −0.4, Tchiral = 0.392±0.0005and TBKT = 0.389 ± 0.0005. Even when comparing with XY model that can be seenas d = −∞ limit for which Tchiral = 0.512 ± 0.0005 and TBKT = 0.506 ± 0.0005, thesplitting looks almost constant. This observation suggests the splitting has little todo with spin fluctuations and consequently with the formation of domain walls but israther mainly due to the screening of vortex-vortex interaction due to domain walls: asa consequence, this screening effect being independent of spin fluctuations, the splittingexhibits little dependence on anisotropy strength d.

Chapter 5

Conclusion

In this doctoral dissertation I have presented a determination of the whole phasediagram of classical Heisenberg triangular antiferromagnet (HTAF) in the strict 2d caseand of its anisotropic derivatives. For the latter ones the case of single-ion anisotropywas chosen for both its experimental relevance and its expected more altering character.In the case of easy-plane anisotropy, the extensive study of XY (XY TAF) model hasappeared necessary as it is the paradigmatic model of the relevant class of universalityand its thorough study was still to be done.

After a statement of the motivations for such a study, I successively presented thecase of HTAF, of easy-axis HTAF and, last, of XY TAF and easy-plane TAF. Howeversurprising it may seem sixty years after the first study of a classical antiferromagneticspin system on the triangular lattice, the phase diagram of the systems here consideredhad not yet been completely built. In case of HTAF and XY TAF, parts of the dia-gram had been discussed but the overall picture was still unsatisfactory and importantfeatures that are not refinements were at best lacking support, at worst fully ignored.

Let’s review the results presented in this dissertation for each system. Zero-fieldbehaviour of HTAF had been studied a lot and with convincing care for me not toundertake another study of it. This system is governed by a continuous symmetry andshouldn’t order at finite temperature as a consequence of Mermin-Wagner theorem.Interestingly Kawamura challenged this viewpoint discussing the topological excita-tions of HTAF and not only showed two temperature regimes for these exist, whichhis opponents agree on, but precisely exposed the reason for the change of regime tobe a transition. His line of arguments parallels the one of Berezinskii, Kosterlitz, andThouless in their discussion of the phase transition now known after their names (BKTtransition): the transition is topological by nature; yet it is distinct of BKT transitionby the decoupling of spin excitations and vortex excitations, which induces a low-temperature phase quite peculiar, coined as spin gel by Kawamura. In any case finite-field behaviour that I studied was astonishingly lacking a precise discussion. The firstsurprising feature of so-far published diagrams was the closing of finite-field transition

59

60 CHAPTER 5. CONCLUSION

lines at a finite-temperature multicritical point which is contradictory with the masslessnature of zero-field phases as opposed to the massive nature of low-temperature finite-field phase. Other important features had already been presented: the existence of anup-up-down collinear phase separating the so-called 120-degree phase from both para-magnetic phase and high-field phase; the selection of a planar high-field phase ratherthan the umbrella structure. A tricky issue had been laid aside, namely the nature ofthe transition at high field between the ordered phase and the saturated/paramagneticphase. A compound S1 ⊗ Z3 symmetry has to be broken. The hardship lies in thefact there is a single order parameter, namely transverse spin component. It appearsthat both symmetries are separately broken, first the discrete Z3 then the continuousS1. Numerical determination of this sequence is somehow delicate as the symmetrydiscussion underpinning this study.

The easy-axis HTAF as presented here in the case of a single-ion anisotropy termexhibits quite rich a zero-field behaviour with triple BKT-like transition. This resultsfrom the specific compound S1 ⊗ Z6 symmetry of the system. The upper two tran-sitions are associated with the breaking of the discrete Z6 symmetry and delimit anintermediate massless critical phase characteristic of Zp models with p > 4 (as p goesto ∞ the lower transition temperature tends to zero to yield the limiting case, namelyXY model). They are BKT-like but not real BKT transitions. The third transition isthe BKT transition associated with the breaking of continuous S1 symmetry. As forfinite-field the behaviour is similar to the one of HTAF as expected by their commonS1 ⊗ Z3 symmetry. In the case of easy-axis HTAF the diagram stretches to highertemperatures thanks to reduced spin fluctuations. Furthermore the anisotropy createsdistinct features: first the splitting of zero-temperature one-third magnetisation tran-sition with a finite field range of existence for the up-up-down collinear phase; secondsaturation field is lowered.

XY TAF presents quite rich a diagram as well. At zero-field it exemplifies the emer-gence of a discrete symmetry in a continuous model as introduced by Villain: here thisdiscrete Z2 symmetry is the well-known chirality. If the successive breaking of Z2 sym-metry and S1 symmetry has been fully demonstrated the precise nature of the uppertransition still excites controversy due to the notorious difficulty to calculate criticalexponents at this transition. In this dissertation I support the viewpoint this transi-tion belongs to Ising class of universality. Finite-field behaviour exhibits interestingpatterns. If Korshunov had carefully discussed them, refinements and numerical evi-dences were necessary. The first feature to be discussed is the existence of a low-fieldmulticritical point that delimits the magnetically disordered chiral phase. Indeed zero-temperature intermediate magnetically disordered chiral phase survive at finite-field upto a multicritical point. It is delimited by transition lines that both emerge from zero-field transitions: the upper one is Z2 symmetry-breaking chiral transition, the lowerone is Z3 symmetry-breaking transition associated with lattice translational symmetry-breaking. The emergence of a Z3 line from a BKT transition is fully allowed in terms of

61

symmetry as Z3 is a subgroup of S1; it is here both supported by numerical results andby the observation that the coupling of spin-waves with field in this region presents athree-fold anisotropy character.At this multicritical point transition lines are inverted:above the multicritical point an intermediate up-up-down collinear phase is stabilisedabove the low-temperature low-field long-range-ordered 120-degree phase. The secondfeature to be put forward is the nature of the transition at high-field into the orderedphase from the paramagnetic/saturated one. The compound Z3⊗Z2 symmetry is bro-ken as a whole Z6 symmetry, which implies the existence of an intermediate masslesscritical phase. The easy-plane HTAF exhibits the same behaviour as XY TAF withoutmuch specificity.

Hence this dissertation sheds light on the behaviour of paradigmatic geometricallyfrustrated magnets. Specifically I have refined finite-field diagram of HTAF and of easy-axis HTAF pointing out a double transition at high field. For XY TAF the diagram isnow precise: I clarify the breaking of emergent Z6 symmetry at high field and the exten-sion to finite field of the magnetically disordered chiral phase. One point still remainsan open question that needs to be assigned, namely very low-field behaviour of HTAF.The dissertation exemplifies the wealth of intricate transitions the intertwining of var-ious discrete symmetries or of discrete symmetries with a continuous one can induce.It also shows that a balanced blend of topological and symmetry arguments with nottoo complicated yet astutely devised numerical simulations can bring much informa-tion and pave the way for more sophisticated machinery to refine details of the overallpicture presented here. This theoretical dissertation naturally finds an experimentalextension: special attention should be given to the following compounds that are quitewell described by models previously discussed. HTAF emblematic system is NiGa2S4

[135]. Yet the most interesting systems are easy-axis HTAF such as Rb4Mn(Mo04)3

[39] or Rb4Fe(MoO4)2 [54].

62 CHAPTER 5. CONCLUSION

Appendix A

Single-ion anisotropy

As this dissertation presents systems in which the interplay between anisotropy andgeometric frustration plays a leading role a short reminder on magnetic anisotropy incrystals doesn’t seem useless (the reader is referred to reference textbooks for detailledcalculations which are not the purpose of this appendix meant as a reminder; see forexample [136, 62]). Magnetic properties of a crystal results from the behaviour ofelectrons in the partially filled outer shells of atoms. Two classes of elements are here ofinterest for their magnetic properties: iron-group transition elements (electrons on 3dshell) on the one hand and rare-earth compounds (electrons on 4f shell) on the otherhand. In both families it is possible to derive a single-ion anisotropy term even thoughthe energy hierarchy is different between both. What are the relevant energetic termsto be considered in order to describe spins in a crystal? There are three terms: crystalfield, spin-orbit coupling LS and Zeeman energy. They respectively read

Vcf (i) = −e∑

j

V (rj) = −e∑

j

1|rj − ri|

(A.1)

with e the electric charge of the electron

VLS = λL · S (A.2)

andVZ = µB(L + 2S) · h (A.3)

where µB is Bohr magneton.In iron-group elements spin-orbit coupling is much weaker than crystal field and

can hence be treated as a perturbation around the structure obtained from crystalfield. Perturbation theory to the second-order for Zeeman and LS terms then yields

H =∑

µν

2µBhν(δµν − λΛµν)Sν − λ2SµΛµνSν − µBhµΛµνhν (A.4)

63

64 APPENDIX A. SINGLE-ION ANISOTROPY

where the tensor (Λµν) is defined as

Λµν =∑

n

〈0|Lµ|n〉〈n|Lν |0〉En − E0

(A.5)

The second term that is the anisotropy term can be developed using x, y, and z crystaldirections as principal axis as

H =13

ÅΛz −

12

(Λx + Λy)ã

(3S2z − S(S + 1)) +

12

(Λx − Λy)(S2x + S2

y) (A.6)

or in shortened notationsDS2z + E(S2

x + S2y) (A.7)

with the constant term dropped.On the contrary in rare-earth elements spin-orbit coupling is the leading interaction.

As a consequence levels first split according to J = L + S values and levels are thensplit by crystal field term. In an hexagonal geometry using the development of crystalfield term with Legendre polynomial on J states let us obtain

Vcf = αA20〈r2〉[3J2z − J(J + 1)]

+ βA40〈r4〉[35J4z − 30J(J + 1)J2

z ]

+ γA60〈r6〉[231J6z − 315J(J + 1)J4

z

+105J2(J + 1)2J2z − 5J3(J + 1)3 + 735J4

z

−525J(J + 1)J2z + 40J2(J + 1)2 + 294J2

z − 60J(J + 1)]

+ γA66〈r6〉16

[(Jx + iJy)6 + (Jx − iJy)6] (A.8)

where α, β, and γ are coefficient depending on J , L, S and l; A20, A40, A60, andA66 are symmetry coefficients in the polynomial expansion. Rare-earth compoundsgenerally exhibit strong anisotropy and the description with the formalism of single-ionanisotropy appears the best suited.

In this work single-ion anisotropy term is used in the form of DS2z which is fully

enough to make non-trivial interplay with geometric frustration appear and to correctlydescribe real compounds (see chap. 6 in [62] and references therein).

Appendix B

Real-space mean-field theory

In this appendix real-space mean-field theory is presented in the case of Heisenbergspin systems. In the first section models with only spin exchange and Zeeman terms aretreated. In the second section models with single-ion anisotropy are carefully discussed.

B.1 Heisenberg Model

Here we treat explicitly an isotropic magnetic system with Heisenberg interactionbetween neighbouring spins.

B.1.1 Single spin in an external field

Let us begin with a single spin in an external magnetic field described by the Hamil-tonian H = −h · S, where S is either a unit classical vector or a quantum spin S. Weneed to calculate the partition function corresponding to the above Hamiltonian.

(i) In the classical case the partition function is given by an integral:

Z =∫ sin θdθdϕ

4πeh cos θ/T =

12

∫ 1

−1dx e−hx/T =

T

hsinhh

T(B.1)

Local magnetization can be obtained by differentiating the free energy F = −T lnZ:

m = 〈cos θ〉 = −∂F∂h

= TZ ′hZ

= cothh

T− Th. (B.2)

(ii) In the quantum case the partition function is obtained by summation of a finiteseries:

Z = Tr e−H/T = ehS/T + eh(S−1)/T + . . .+ e−hS/T =sinh h(S + 1

2)/T

sinh h/2T. (B.3)

65

66 APPENDIX B. REAL-SPACE MEAN-FIELD THEORY

The magnetization is expressed as

m =ÅS +

12

ãcoth

h(S + 12)

T− 1

2coth

h

2T. (B.4)

B.1.2 Lattice model

Consider the following spin Hamiltonian:

H =∑

〈ij〉Jij Si · Sj − h ·

∑

i

Si . (B.5)

The mean field approach amounts to neglecting correlations between spin fluctuationson adjacent sites:

Si · Sj =îmi + (Si −mi)

ó·îmj + (Sj −mj)

ó

≈ mi ·mj + mj · (Si −mi) + mi · (Sj −mj)

= Si ·mj + Sj ·mi −mi ·mj (B.6)

where mi = 〈Si〉 is a thermodynamic average of an on-site spin. The mean fieldHamiltonian, then, is

HMF = −∑

〈ij〉Jijmi ·mj −

∑

i

hi · Si , (B.7)

wherehi = H−

∑

j

Jijmj . (B.8)

The problem of interacting spins is, thus, reduced to a set of single-spin problemssupplied by the self-consistency equations:

mi =hihi

ïcoth

hiT− Thi

òor mi =

hihi

ïÅS+

12

ãcoth

hi(S + 12)

T− 1

2coth

hi2T

ò(B.9)

with local fields defined by Eq. (B.8).After the solution of self-consistent equations is found the internal energy EMF and

the free energy FMF are calculated by

EMF =∑

〈ij〉Jijmi ·mj−

∑

i

hi ·mi and FMF = −∑

〈ij〉Jijmi ·mj−T

∑

i

lnZi , (B.10)

where Zi is given by Eqs. (B.1) or (B.3) depending on the nature of spins.

B.2. ANISOTROPIC MODEL 67

B.2 Anisotropic Model

The above expressions with very little modifications can be transferred to anisotropicsystems with anisotropy coming from interspin interactions (anisotropic exchange,Dzyaloshinskii-Moria interaction, dipolar interactions). In the presence of single-ionanisotropy the basic equations must be modified significantly. Below we treat explicitlythe case of classical spins.

B.2.1 Classical spins problem

The single-site Hamiltonian is

H = D(n · S)2 − h · S (B.11)

where D is a single-ion anisotropy constant and n is an anisotropy axis. We split localmagnetic field into two parts: component parallel to anisotropy axis h‖ = h · n andtransverse component: h⊥ =

»h2 − (h‖)2. In the classical case the partition function

is given by

Z =∫ sin θdθdϕ

4πe−E(θ,ϕ)/T , E(θ, ϕ) = D cos2θ − h‖ cos θ − h⊥ sin θ cosϕ (B.12)

while the resulting magnetization has two components in the direction parallel andperpendicular to the axis n:

m = m‖n +m⊥îh− n(h · n)

ó/h⊥

m‖ =1Z

∫ sin θdθdϕ4π

cos θ e−E(θ,ϕ)/T (B.13)

m⊥ =1Z

∫ sin θdθdϕ4π

sin θ cosϕ e−E(θ,ϕ)/T

Partial integration over azimuthal angle simplifies Eqs.. (B.12) and (B.13) to

Z =12

∫ 1

−1dx e−Dx

2/T eh‖x/T I0(y)

m‖ =1

2Z

∫ 1

−1xdx e−Dx

2/T eh‖x/T I0(y) (B.14)

m⊥ =1

2Z

∫ 1

−1dx√

1− x2e−Dx2/T eh

‖x/T I1(y)

where y = (h⊥√

1− x2)/T and I0(z) and I1(z) are the modified Bessel functions ofzeroth and first order:

I0(z) =1π

∫ π

0ez cosϕdϕ , I1(z) =

1π

∫ π

0ez cosϕ cosϕdϕ . (B.15)

68 APPENDIX B. REAL-SPACE MEAN-FIELD THEORY

The lattice Hamiltonian is written as

H = D∑

i

(ni · Si)2 +∑

〈ij〉Jij Si · Sj − h ·

∑

i

Si , (B.16)

where the anisotropy axis can be either uniform or staggered. The mean-field approxi-mation leads to the following Hamiltonian:

HMF = −∑

〈ij〉Jijmi ·mj +D

∑

i

(ni · Si)2 −∑

i

hi · Si , hi = h−∑

j

Jijmj (B.17)

supplied by the mean-field equations (B.14). The free-energy is still given by the aboveEq. (B.10), while an expression for the internal energy becomes

EMF =∑

〈ij〉Jijmi ·mj −

∑

i

hi ·mi +D∑

i

〈(m‖i )2〉 . (B.18)

Here, square of the parallel component of the on-site moment is given by

〈(m‖)2〉 =1

2Z

∫ 1

−1x2dx e−Dx

2/T eh‖x/T I0(y) . (B.19)

B.2.2 Quantum case S=1

Let’s now introduce the case of a quantum system. Single-ion anisotropy is relevantfor S > 1 as (S · n)2 = const in case of S = 1/2. Provided we consider the anisotropyaxis along z the single-site anisotropy can be written

H = D(Sz)2 − h‖Sz − h⊥S+ + S−

2(B.20)

In the case of spin S = 1 this can be written as

H =

áh‖ +D −h⊥√

20

−h⊥√2

0 −h⊥√2

0 −h⊥√2−h‖ +D

ë

(B.21)

Introducing the characteristic polynomial the eigenvalues can be calculated and themagnetisation deduced

χH(E) = −E3 + 2DE2 + (h2 −D2)E −D(h⊥)2 (B.22)

Ek = 2

−p3

cos

(

13

arccosÅ−q

2

√

27−p3

ã+

2kπ3

)

− 2D3

(B.23)

B.2. ANISOTROPIC MODEL 69

where p = −(D2/3 + h2) and q = 2D3/27 + D(−2(h‖)2/3 + (h⊥)2/3). Then usingm‖,⊥ = −T∂h‖,⊥ lnZ and ∂h‖,⊥χH(Ek) = 0

Z =∑

k

e−Ek/T , m‖ =1Z

∑

k

−2h‖Eke−Ek/T

3E2k − 4DEk +D2 − h2

, m⊥ =1Z

∑

k

−(2Ek −D)h⊥e−Ek/T

3E2k − 4DEk +D2 − h2

(B.24)Using the mean field approximation the lattice Hamiltonian can be expressed as in(B.17), which enables to use the above results for the single-site.

70 APPENDIX B. REAL-SPACE MEAN-FIELD THEORY

Appendix C

Zero-field upper transition inmean-field treatment: sign of γ2

In this appendix the sign of one of the sixth-order coefficients in Landau-Ginzburgfunctional about zero-field upper transition is determined using mean-field description.For the determination of some phenomenological parameters it is necessary to use thedescription of a layered system and consider the specific case of fully decoupled layersto obtain the sign of γ2 in case of the purely 2d system.

Let’s recall mean-field equations:

m⊥i =1

2Zi

∫ 1

−1dx√

1− x2 eDx2/T eh

zix/T I1(yi) (C.1)

mzi =1

2Zi

∫ 1

−1dx x eDx

2/T ehzix/T I0(yi) (C.2)

Zi =12

∫ 1

−1dx eDx

2/T ehzix/T I0(yi) (C.3)

where yi = h⊥i√

1− x2/T , hi = h−∑n.n.mj, and In(z) is the modified Bessel functionof the n-th order:

In(z) =1π

∫ π

0dα ez cosα cosnα . (C.4)

At the upper transition from paramagnetic phase to collinear one h⊥i = 0, whichimplies yi = 0, and m⊥i = 0 as well: Eq. (C.1) can be dropped. For clarity we hereafterdrop the index z: hzi = hi, etc. Let’s expand Eqs. (C.3) and (C.2) in powers of hiaround hi = 0. For calculations to remain legible abridged notations are introduced asfollows:

Kn =∫ 1

−1dx xne

dx2

T ∀n ∈ N, K2n+1 = 0 (C.5)

71

72 APPENDIX C. SIGN OF γ2

For our purpose following integrals will be used:

K0 =

√πerfi

(»d/T

)

»d/T

K2 =ed/T

d/T− K0

2d/T

K4 =2d/T − 32(d/T )2

ed/T +3K0

4(d/T )2

K6 =15− 10d/T + 4(d/T )2

4(d/T )3ed/T − 15

8(d/T )3K0

where erfi designates the imaginary error function defined as

erfi(z) =erf(iz)i

with erf(z) =2√π

∫ z

0du e−u

2

(C.6)

With these notations the expansion to fourth-order of Eq. (C.3) reads

Zi =K0

2

ñ1 +

K2

2K0T 2h2i +

K4

24K0T 4h4i

ô+O(h6

i ) (C.7)

thus1Zi

=1Zi

ñ1− K2

2K0T 2h2i +

6K22 −K0K4

24K20T

4h4i

ô+O(h6

i ) (C.8)

To calculate mi, let’s start with the calculation of the integral in Eq. (C.2):

∫ 1

−1dx xe

dx2+hix

T =K2

T+K4

6T 3h3i +

K6

120T 5h5i +O(h7

i ) (C.9)

Then combining with Eq. (C.8) Eq. (C.2) becomes

mi = b1hi + b3h3i + b5h5

i +O(h7i ) (C.10)

where

b1 =K2

K0T(C.11)

b3 =K0K4 − 3K2

2

6K20T

3(C.12)

b5 =K6K

20 − 15K0K2K4 + 30K2

2

120K30T

5(C.13)

73

Now self-consistent equations are derived in the context of layered systems with aninterlayer coupling λ, with λ > 0 for antiferromagnetic coupling and λ < 0 for ferro-magnetic coupling. Let’s write down the definition of local fields in the case of an an-tiferromagnetic coupling (six-sublattice structure: m1,m2,m3 on one layer, m′1,m

′2,m

′3

on the other):

m1 = l1m2 = −l1−

√3l2

2

m3 = −l1+√

3l22

m′1 = −l1m′2 = l1+

√3l2

2

m′3 = l1−√

3l22

(C.14)

and

h1 = −3(m2 +m3)− 2λm′1 = (3 + 2λ)l1h2 = −3(m1 +m3)− 2λm′2 = −(3 + 2λ) l1+

√3l2

2

h3 = −3(m1 +m2)− 2λm′3 = −(3 + 2λ) l1−√

3l22

h1 = −(3 + 2λ)l1h2 = (3 + 2λ) l1+

√3l2

2

h3 = (3 + 2λ) l1−√

3l22

(C.15)where antiferromagnetic order parameters have been introduced:

mi = l1 cos(q0 · ri) + l2 sin(q0 · ri) (C.16)

As a consequence solving Eq. (C.10) is equivalent to solving the following polynomialequation: Ä

b1 − 1 + b3x2 + b5x4äx = 0 (C.17)

where the term O(x7) has been dropped, x ∈ l1, (l1 ±√

3l2)/2, and renormalisedcoefficients bi’s read (one can easily check that the treatment of ferromagnetic andantiferromagnetic couplings are equivalent, hence the introduction of |λ| to encompassboth situations in the following)

bk = (3 + 2|λ|)kbk (C.18)

Solutions to Eq. (C.17) are either x = 0 or solutions toÄb1 − 1 + b3x2 + b5x4

äthat can

be written as

y± =−b3 ±

√B

2(3 + 2|λ|)2b5(C.19)

where

B = b23 − 4Çb1 −

13 + 2|λ|

åb5 (C.20)

Then solutions can be expressed in terms of l1 and l2:

l1 = l2 = 0 (C.21)

l1 = ε√y+

l2 = 1√3ε√y+

(C.22)

74 APPENDIX C. SIGN OF γ2

l1 = ε√y−

l2 = 1√3ε√y−

(C.23)

where ε = ±1.Of course conditions of positivity for B and y± have been taken into account: on

the temperature range [1.04, 1.30] B > 0 y± > 0.Let’s now turn to Landau-Ginzburg free energy functional:

F = α(l21 + l22) + β12(l21 + l22)2 + γ1(l21 + l22)3 + γ2[(l1 + il2)6 + cc] (C.24)

In this expression there is only a single fourth-order term as the development is madeabout a collinear phase in which case both fourth-order terms have the same expression.At equilibrium ®

∂l1F = 0∂l2F = 0

(C.25)

which yields®α+ 2β12(l21 + l22) + 3(γ1 − 4γ2)l41 + 3(γ1 + 4γ2)l42 + 6γ1l21l

22 = 0

3γ2l41 + 3γ2l42 − 10γ2l21l22 = 0

(C.26)

In terms of l1 and l2 solutions read:

l1 = l2 = 0 (C.27)

or

l1 = 0 and l2 = ±

Ã−β12 ±

√δ′

3(γ1 − 2γ2)(C.28)

or

l1 = ±

Ã−β12 ±

√δ

3(γ1 + 2γ2)and l2 = 0 (C.29)

or

l1 = ±

Ã−β12 ±

√δ

12(γ1 + 2γ2)and l2 = ±

Ã−β12 ±

√δ

4(γ1 + 2γ2)(C.30)

or

l1 = ±

Ã−β12 ±

√δ

4(γ1 − 2γ2)and l2 = ±

Ã−β12 ±

√δ

12(γ1 − 2γ2)(C.31)

where δ = β212 − 3α(γ1 − 2γ2) and δ′ = β2

12 − 3α(γ1 − 2γ2).From the comparison of both sets of solutions it is then possible to extract an

expression for γ2, which is the purpose of these calculations:

γ2 =b54

Ç4

3α

Çb1 −

13 + 2|λ|

å− 1

2(3 + 2|λ|)2

å(C.32)

Taking |λ| = 0 in Eq. (C.32) yields γ2 > 0 in the vicinity of the upper transition atzero field, Tc1 ≈ 1.2 QED.

Appendix D

Finite-size scaling

A striking paradox lies at the very heart of our experience of phase transitions:trusting our senses they do occur in systems around us, trusting our reason they aremathematically possible only for infinite systems. As often physics just stands in be-tween as shows the introduction of the thermodynamic limit, which can be understoodas a way to take an infinite limit to describe physical systems that are necessarily fi-nite. Yet such a solution encounters its limits of accuracy as we start dealing withsystems characterised by small extent in one or several dimensions. Systems used innumerical works do exhibit strong alterations to thermodynamic behaviour as they areoften of very limited sizes. Then it is necessary to question the way thermodynamiclimit is approached and what the correction terms are. In the late 1960’s Fisher initi-ated the answering to this concern. If I don’t aim at reproducing this work hereafter,which is quite clearly exposed in original papers [31, 32, 30] and the textbook review byBarber [8], I will remind most important assumptions that lay the foundation for thisanalysis of correction terms for finite-size systems, otherwise called finite-size scaling.Interestingly what could be perceived a hindrance to numerical investigation of phasetransitions has in fact been turned into the way to extract information on phase tran-sitions from calculations on finite-size systems. Of course it doesn’t mean misleadingresults in finite-size clusters can always be circumvented.

D.1 Fundamentals

D.1.1 Geometry and boundary conditions

First the system under consideration has to be precised, especially the boundaryconditions that are adopted. There are three different geometries dealt with in scalingtheories:

• G1: A completely finite system of volume V = Ld in d dimensions (d′ = 0)

75

76 APPENDIX D. FINITE-SIZE SCALING

• G2: A d-dimensional layer of infinite dimension in d′ = d − 1 dimensions and offinite thickness L

• G3: A system finite in d− 1 dimensions and infinite in the last one with a cross-section area characterized by length L (d′ = 1)

G1 corresponds to the situation of real systems and clusters used in Monte Carlo nu-merical investigations as presented in this work. G2 is of interest mainly for analyticalcalculations and enable to study dimensional crossover phenomena. G3 is the geome-try used in transfer matrix method and phenomenological renormalisation group or inquantum systems, with time as the infinite dimension.

When considering finite systems boundary conditions have to be assigned: these canbe expressed using a function ϕ defined on a d′ × (d − d′) space (the first componentcorresponds to dimensions along which the system is infinite, the second to those alongwhich it is finite) with (ei)16i6d−d′ a basis of the second component:

• Periodic boundary conditions: ∀i, u, v, ϕ(u,v + Lei) = ϕ(u,v)

• Open boundary conditions (or free surface): ∀u, v, ϕ(u,v) = 0 if v /∈ [0, L]d

• Twisted boundary conditions, among which antiperiodic ones: ∀i, u, v, ϕ(u,v+Lei) = −ϕ(u,v)

Each set of conditions presents advantages depending on the use: for example the latterones are often introduced in the study of helicity modulus, aka stiffness. Open boundaryconditions may be helpful when carrying out analytical calculations. The reason whyperiodic boundary conditions are used in most numerical works is the quick convergenceof functions towards thermodynamic limit. Let’s indeed assume thermodynamic limitexists:

f∞(T, ρ) = limN→∞V→∞

ρ=N/V=const

F (T, V,N)V

(D.1)

where F is the free energy, T the temperature, V the volume, N the number of particlesand ρ the density. Then free surface boundary conditions yield surface terms:

F (T, V,N) = V f∞(T, ρ) + Afx(T, ρ) + o(A) (D.2)

where A designates the area of the boundary. Periodic boundary conditions induce anexponential convergence:

F (T, V,N) = V f∞(T, ρ) +OÄe−LΓ(T )

ä(D.3)

This expansion however breaks down in the vicinity of critical temperature Tc aslimT→Tc

Γ(T ) = 0, which explains the so-called critical slowing-down about phase tran-

sition in numerical calculations.

D.1. FUNDAMENTALS 77

D.1.2 Alteration of singularities in finite systems

As previously seen finite size induces corrections to thermodynamic functions and es-pecially a rounding of singularities. Let’s consider a diverging thermodynamic quantityand discuss the behaviour of its estimate in finite-size systems for which no singularityappears (typically in geometry G1 that is relevant for Monte Carlo numerical works).Its estimate on finite systems exhibits a peak instead of a divergence, the height ofwhich increases with l = L/a (a is a typical microscopic lengthscale of the system),located at Tm(l) that tends to Tc, the thermodynamic critical temperature, as [31, 32]

Tm(l)− TcTc

∼ blλ

(D.4)

Another temperature of interest is the temperature T ∗(l) at which the quantity eval-uated on a finite-size system starts departing significantly from the thermodynamicfunction. T ∗ tends to Tc as [31, 32]

T ∗(l)− TcTc

∼ clθ

(D.5)

Fisher and Ferdinand put forward the following argument to assign a value to θ [31]:T ∗ is such that ξ(T ∗) = L. In other words once the correlation length spans the wholesystem it is no longer possible for criticality to grow. This argument yields θ = 1/ν asξ(T ) ∼ (T −Tc)−ν . For λ the situation is less clear; yet a similar line of reasoning leadsto λ = 1/ν.

D.1.3 Scaling

After having described the rounding of singularities in finite systems, next point isto explain how to nevertheless extract information on thermodynamic singularity, ie ontransition, from finite-size data. This is the aim of finite-size scaling. Finite-size scalingis based on the following Ansatz [32, 30]: in the vicinity of the bulk critical temperatureTc the behaviour of a system with at least one large but finite dimension is determinedby

y =L

ξ(T )(D.6)

with ξ the bulk correlation length. Then considering a thermodynamic quantity PL(T )of the finite system with an algebraic singularity in the infinite system P∞ ∼

t→0c∞t

−ρ

where t = (T − Tc)/Tc is the reduced temperature, the finite-size scaling hypothesisreads

PL(T ) ∼ lωRP (y) (D.7)

78 APPENDIX D. FINITE-SIZE SCALING

or in terms of the reduced temperature as can be derived transforming scaling variable(D.6) introducing the algebraic divergence of ξ (ξ(T ) ∼ (T − Tc)−ν)

PL(T ) ∼ lωQP (lθ t) (D.8)

where t = T − Tc(L) and Tc(L) is a pseudo-critical temperature – in fact as long asthe shift exponent λ > 1/ν there is a certain flexibility in the definition of this pseudo-critical temperature based either on T ∗ or Tm; it is even possible to substitute t = T−Tcto t in certain formulas as done in the following section where this formalism is actuallyused. Then for these definitions to be coherent and reconcile with the infinite limitexponents have to respect

ω = ρθ =ρ

ν(D.9)

In the case here considered (no transition in the finite system) limx→0QP (x) = Q0 so that

Pl(Tc(l)) ∼l→∞Q0lρ/ν (D.10)

This relation states that the way estimates of thermodynamic quantities vary withsystem size depends on thermodynamic critical exponents. It constitutes the heart offinite-size scaling techniques to estimate critical exponents and critical temperature.

The preceding calculations are applicable in case of an algebraic singularity in theinfinite system. In case the thermodynamic quantity under scrutiny exhibits a loga-rithmic divergence as is the case of specific heat in bidimensional Ising model:

P∞(T ) ∼ c∞ ln t (D.11)

then Eq. (D.8) cannot be used. Scaling is however possible introducing a non-criticaltemperature T0 and then write [32]

(PL(T )− PL(T0)) ∼ÄQP (lθt)−QP (lθt0)

ä(D.12)

Matching limiting cases yields once again the scaling equation of practical use:

PL(Tc(L)) ∼l→∞t→0

−c∞ν

ln l +O(1) (D.13)

D.2 Expressions of direct interest in this study

This section aims at summarizing the quantities used in the numerical calculationsof this work. It starts with a specification of the general expressions introduced in theprevious section before a discussion of other interesting probes to locate and characterisetransitions, namely Binder’s cumulant and a ratio of correlation functions astutely

D.2. EXPRESSIONS OF DIRECT INTEREST IN THIS STUDY 79

chosen. The way procedures to locate transitions based on these equations can be builtis extensively discussed.

In preceding section as already mentioned some results can be expressed in terms ofthe critical temperature of the infinite system. In particular (D.10) can be written interms of Tc. Hence for specific heat the exponent ρ is α; for susceptibility (D.10) reads

χ(Tc, l) ∼l→∞Q0l

2−η (D.14)

where γ/ν = 2 − η, when d = 2, has been used. This has been extensively used tolocate Tc as it shows that χ(Tc, l)l−(2−η) is size independent: in other words Tc is thetemperature of the intersection points of plots of χ(Tc, l)l−(2−η) versus temperature fordifferent cluster sizes.

Another consequence of scaling was drawn by Binder when he introduced the cumu-lant now often referred to after his name [12]. For the purpose of scaling scalar factorsintroduced in the original paper can be taken off:

UA(T, L) =〈A4〉〈A2〉2 (D.15)

where A is the order parameter of interest (eg: Sαq with α ∈ x, y, z, chirality κ, etc.).Applying scaling to fourth and second moments that appear in this definition yieldsthe following

UA(T, L) = fA(y) (D.16)

A straightforward and important consequence of this expression is that UA(Tc, L) issize-independent (fA(y) −→

y→0f 0A): plots of Binder’s cumulant versus temperature for

different L cross at Tc in case of a second-order transition, merge in case of a BKTtransition. Another consequence is that it can be used to estimate critical exponents[70]. Let’s illustrate the method with susceptibility: expression (D.7) used for χA (whereA is the order parameter of interest) shows that χA/L2−η is a function of y; given (D.16)it results that plotting χA/L2−η versus UA with η as a free parameter can be used toestimate η: for the right value plots are indeed L-independent, which means all datapoints collapse on a single curve.

An improvement has recently been brought with the consideration of a ratio ofcorrelation functions [120, 112] which has similar properties to Binder’s cumulant withthe great advantage of better accuracy as it is a second-order quantity. Let’s considerspin-spin correlation function

g(r, L, T ) =1Ld

∑

i

〈Si · Si+r〉 (D.17)

where the abridged notation i + r indexes neighbours of spin i at distance r. Scalingform of g introduces another length ratio beside y = L/ξ:

g(r, L, T ) ∼ r−(d−2+η)QÅ rL, yã

(D.18)

80 APPENDIX D. FINITE-SIZE SCALING

As a consequence

Γ =g(L/2, L, T )g(L/4, L, T )

=ÇL/2L/4

å−(d−2+η) Q(

L/2L, y)

Q(

L/4L, y) = Q(y) (D.19)

As it is a second-order expression statistical noise affects less severely this quantity. Itcan then be used in a similar way to Binder’s cumulant and can also help identifyingan intermediate algebraic phase that is otherwise invisible using Binder’s cumulant.

Last, the case of BKT transition has to be addressed. In numerical calculations oneof the most efficient ways to study a BKT transition is spin stiffness ρS (see Eq. 2.23).At a BKT transition spin stiffness exhibits a jump the height of which is universal,equal to ρS(TBKT ) = (2q2/π)TBKT for a transition driven by 1/q vortices. For integervortices (q = 1) it thus reads ρS(TBKT ) = (2/π)TBKT . Renormalisation group flowyields the following scaling for spin stiffness [127, 94]:

ρS(T, L) = ρ∞S (T )Ç

1 +1

2(lnL+ c)

å(D.20)

From this formula a way to extrapolate TBKT can be devised: if TL is such thatρS(TL, L) = (2/π)TL then

TL − TBKTTBKT

=1

2(lnL+ c)(D.21)

Often an approximate expansion in terms of 1/L can be used instead due to negligiblecorrection terms.

Appendix E

Monte Carlo algorithms

To study thermal averages in equilibrium statistical mechanics a powerful methodarose with the development of numerical simulations, viz. Monte Carlo simulation: ituses a stochastic trajectory to explore the phase space. This method is quite efficientas it enables to study any classical system even if analytically intractable and offersresults that can often be compared with experiments. Concerning quantum systemsthe situation is less favourable due to the notorious sign problem. Quantum MonteCarlo techniques are quite distinct of classical Monte Carlo and won’t be dealt withhereafter: only classical Monte Carlo algorithms are indeed of interest for the presentwork.

This appendix is not meant to provide an extensive presentation on Monte Carloalgorithms, as may be found in [13, 45], but to review the basics about classical MonteCarlo.

E.1 Back to basics

Let’s consider a system with Hamiltonian H. We are interested in calculating suchproperties of this system as 〈A〉T (canonical ensemble) where A is an observable (internalenergy, magnetisation, etc.) and the average is a thermal one. In other words we wantto average an observable over phase space Ω respecting a certain distribution of statesω in this space (typically ω(x) = exp [−βH(x)]):

〈A〉T =∫

Ω dx A(x)ω(x)∫

Ω dx ω(x)(E.1)

For such an average to be calculated an efficient integrating algorithm has to be used:in high dimensions the most efficient one is a Monte Carlo integration. Monte Carloalgorithms are based on the use of a pseudo-random variable: most of the time it isenough to use variables provided by a well-designed random number generator rather

81

82 APPENDIX E. MONTE CARLO ALGORITHMS

than using a list of ‘true’ random numbers obtained, eg, through nuclear radiations andthus we won’t make any distinction in the following. In our work we implemented arandom number generator proposed by Marsaglia [72].

If phase space were to be explored in a fully random manner the number of MonteCarlo steps required for a good accuracy would be awfully huge as the distributionof states in the phase space appears to be too peaked to make a simple samplingmethod efficient. It is thus most desirable to devise an importance sampling that couldpreferably sample regions according to the equilibrium distribution ω: a Markov chainoffers such a possibility in a way perfectly adapted for numerical calculations as it hasno memory: the evolution only depends on the current state and the transition matrixT defining the chain. Let’s remind what a Markov chain is and review the conditionsfor a Markov process to converge to a given distribution ω. For simplicity we hereafterpresent the case of a Markov chain in a finite space : going to infinite discrete or evencontinuous limit can be done in a natural way for spaces relevant to physical systems.

First let’s recall what a stochastic matrix is: let T be a matrix; then T is said to be astochastic matrix if all its row vectors are probability distributions. A Markov chain is astochastic process defined as a sequence of random variables (Xn)n>1 on a space S suchthat P (Xn+1 = i|Xn = j) = Tij with (Tij) a stochastic matrix. To initiate the process,an initial distribution λ is chosen. We note (X,λ, T ) such a Markov process. For ourpurpose the initial distribution λ doesn’t matter as we aim at gaining information onthe stationary distribution π of the process, which is such that Tπ = π and that we wantverifying π = ω. The existence of a stationary distribution is guaranteed by the ergodictheorem: let (X,λ, T ) be a Markov process over a finite space S with T an aperiodicand irreducible stochastic matrix, then there exists a unique stationary distribution π.Furthermore, for any initial distribution λ and bounded function f over S, then

limN→∞

1N

∑

i∈(Xn)Nn=1

f(i) =∑

i∈Sπif(i) almost surely (E.2)

An aperiodic and irreducible transition matrix for a Markov process prevents any finiteperiod of return for all states after a certain rank in the process and assures that anystate is reachable from any other state during the process. As a consequence, the chaincan be assumed independent of the initial distribution λ after a certain rank dependingon its speed of convergence towards its stationary distribution. λ-dependent initialsteps are typically called thermalisation steps and are discarded for measurement.

The next question is how to construct the transition matrix T such that its station-ary distribution is the given distribution π = ω. There are various ways of dealing withthis problem: we will concentrate on processes that can be described with a transitionmatrix reading:

Tij = PijAij + δij∑

k

Pkj(1− Akj) (E.3)

where P is a proposal matrix and A an acceptance matrix. Time can be introduced

E.1. BACK TO BASICS 83

in such a Markov chain through a characteristic time τ0 associated with an elementarystep, multiplying Tij by τ−1

0 . Provided P is a symmetric stochastic matrix and A, amatrix with elements in [0, 1], respecting detailed balance with respect to π:

∀(i, j), Aijπj = Aijπi (E.4)

then the Markov chain associated with T admits π as its stationary distribution andproperty (E.2) applies. Different choices then exist for both P and A [45, 71]. Let’s firstdiscuss the choice of the acceptance matrix A. Among the most widely used methodsare Hastings’ ones [38]:

Aij = f(i, j) minÇπjf(j, i)πif(i, j)

, 1å

(E.5)

where f is an appropriate auxiliary function. The well-known algorithm proposed byMetropolis et al. [78] corresponds to f = 1 in (E.5) with Boltzmann’s distribution asπ.

Various proposal matrices have been devised. In the case of non-frustrated systemscluster algorithm are certainly among the most efficient ones; alas in case of the frus-trated systems considered in this work they cannot be used. Therefore we present herethe solution we have adopted to update continuous Heisenberg or XY spins, namely asingle-spin restricted motion scheme.

A naïve approach for Heisenberg spins consists in generating a random vectoruniformly distributed on S2. This can be achieved using an angular representation:dz = sin(θ)dθ and dφ are uniform distributions. Then one can proceed as follows:

z = p1

phi = p2*pi

r = sqrt(1 - p1**2)

x = r cos(phi)

y = r sin(phi)

where p1 and p2 are random numbers uniformly distributed in [−1, 1] obtained beforethis sequence by calling a random number generator. This method is quite heavyin calculations as it calls two trigonometric functions and one root square. Anotherstraightforward method is a von Neumann rejection procedure:

(1) z = p1

y = p2

x = p3

r = sqrt(x**2+y**2+z**2)

if(r.gt.1) then goto (1)

x = x/r

y = y/r

z = z/r

84 APPENDIX E. MONTE CARLO ALGORITHMS

This is also quite CPU consuming as it uses a square root and has a poor acceptanceratio a: a ≈ Vsphere/Vcube ≈ 52%.

Marsaglia proposed another approach based on the fact that a uniform angular dis-tribution (α, β) also yields a uniform distribution of the radius α2+β2. As a consequenceif (α, β) is uniformly drawn in [−1, 1]× [−1, 1], a rejection method then yield a uniformdistribution on the unit-radius circle with an acceptance ratio a ≈ Adisc/Asquare ≈ 72%.This uniformly distributed random variable α2 + β2 ∈ [0, 1] is then used to build z. xand y are proportional to α and β respectively: one has just to renormalise the lattervariables. The algorithm reads:

(1) random(p1)

random(p2)

if((p1**2+p2**2).gt.1.d0) goto (1)

z = 1 - 2*(p1**2 + p2**2)

a = sqrt(2*(1+z))

x = a*p1

y = a*p2

Landau and Binder [64] introduced in the context of the unfrustrated anisotropicHeisenberg model on the square lattice a method to avoid the dramatic plummetingof acceptance ratio at low temperature, namely restricted motion. They base thisproposition on the fact that nearby configurations are likely to have close-by energies.This method can be adapted to frustrated systems and to the specific proposal generatorwe use. This improvement devised by Zhitomirsky can easily be understood in terms oflocal coordinates: sampling in a small cone about the spin to be updated is equivalent toa reduction of the interval of distribution of z which amounts to replace z = 1−2(α2+β2)by z = 1− δs(α2 + β2) with δs ∈]0, 2[ adjusted to keep an acceptable acceptance rate.Such a restriction restricts the dynamics in phase space, which may be a matter ofconcern when phase space is very craggy: it is then possible to be locally trapped outof equilibrium in a metastable configuration.

For XY spins to avoid the use of trigonometric functions one can use two randomvariables for cartesian coordinates; for restricted motion to be implemented no suchequivalent of what is shown above for Heisenberg spins is known to us: at most consid-ering local framework proposals can be restricted to the half plane in which the spin ispointing.

E.2 Out of traps: over-relaxation

When dealing with systems like frustrated magnets it is quite common to encounterthe problem of trapping as previously described. Unfortunately most clever updatingschemes designed for simple systems such as Ising ferromagnet don’t work for frustrated

E.2. OUT OF TRAPS: OVER-RELAXATION 85

magnets. A way out is provided by the mixing with quite different a technique, namelyover-relaxation. It consists in proposing a new state the action of which is not increasingas compared with the current one; the new state is related to the current one by a sym-plectic transformation [24]. This evolution can be made micro-canonical as proposedhereafter. For simplicity when we refer to over-relaxation we mean micro-canonicalover-relaxation.

Let’s illustrate how over-relaxation works in Heisenberg spin systems. In case themodel can be written in terms of a local field:

H =∑

i

hi · Si (E.6)

where hi is a function independent of Si, any rotation of Si around hi is a micro-canonical evolution. The most efficient rotation to calculate thermal averages is πrotation around hi [24, 42, 43]:

S′i = −Si + 2S‖i

= −Si + 2hihi · Sih2i

(E.7)

Such an evolution enables the evolution to a far-away state in phase space enhancingergodicity, hence convergence. Furthermore this is a deterministic step as it doesn’trequire any random number. In case of Heisenberg models with a single-ion anisotropyterm, two different terms control the evolution of a given spin: the local field and theanisotropic term. Let’s illustrate what a micro-canonical evolution looks like in thiscase.

H =∑

i

Ähi · Si − d(n · Si)2

ä(E.8)

To keep both terms constant in (E.8), a possible action is to reflect Si with respect tothe plane (n,hi):

S′i = Si − 2S⊥i

= Si − 2(n× hi) · Si

(n× hi)2 (n× hi) (E.9)

If investigations using pure over-relaxation algorithms exist (eg. [42]), which requirescareful treatment of the updating process in order to obtain an ergodic evolution, it isoften interesting to combine over-relaxation into Monte Carlo so that it is still possibleto work in the canonical ensemble that presents some advantages eg in the use offinite-size scaling techniques. A typical mixed algorithm consists in the insertion of anover-relaxation step after each Monte Carlo step. In our work an over-relaxation stepwas devised as a sequential single-spin change according to (E.7) or (E.9) dependingon the case. In a pure over-relaxation algorithm sequential updating has to be avoidedfor ergodicity considerations; in our algorithm as Monte Carlo steps breaks sequentialupdating it is not a matter of concern.

86 APPENDIX E. MONTE CARLO ALGORITHMS

E.3 Estimating errors

As in any numerical work it is necessary to estimate errors on the obtained estimates.Two kinds of errors have to be considered [13]: statistical errors and controllable system-atic errors. Concerning the latter, once errors due to a bad random number generatorhave been removed, there remains those due to finite-size clusters and finite-lengthMarkov chains. As explained in Appendix D, the first group of errors can be turnedinto a powerful source of information; regarding the second group one can interpret theMarkov chain as a dynamical process: one can then discuss of relevant parameters af-fecting the error as further explained below. Let’s consider N successive measurementsof A. The expectation value of the squared error can be written [13]:

¨(δA)2

∂= 〈[ 1N

N∑

µ=1

(Aµ − 〈A〉)ó2∂

(E.10)

where the abridged notation Aµ = A(Sµ) is used.The handling of errors is not as trivial as in the case of the average of independent

random variables. In Markov chains, states are highly correlated. Let’s introduce theautocorrelation time τA that indicates the typical time over which configurations arecorrelated. Shifting to a time representation with t = δt µ where δt = nτ0, τA can bedefined as follows:

τA =∫ ∞

0dt ϕA(t) (E.11)

with

ϕA(t) =〈A(0)A(t)〉 − 〈A〉2〈A2〉 − 〈A〉2 (E.12)

Then (E.10) can be written:

¨(δA)2

∂=

1N

Ä〈A2〉 − 〈A〉2

ä ®1 +

2δt

∫ tN

0dt

Ç1− ttN

åϕA(t)

´(E.13)

where tN = Nδt. If tN ≫ τA then 〈(δA)2〉 reads:

1 + 2τA/δtN

Ä〈A2〉 − 〈A〉2

ä(E.14)

If δt ≫ τA ie n ≫ τA/τ0 (in other words measurements are carried out every nelementary steps with n much larger than the autocorrelation for the observable Aacting on the states of the Markov chain), autocorrelations can be neglected, and theerror on the average can be estimated by the standard formula:

¨(δA)2

∂≈ 〈A2〉 − 〈A〉2 (E.15)

E.3. ESTIMATING ERRORS 87

On the contrary if δt≪ τA the error reads:

¨(δA)2

∂≈ 2τAtN

Ä〈A2〉 − 〈A〉2

ä(E.16)

In this case the only relevant parameter is τA/tN which means either the length ofthe Markov chain as to be improved or τA reduced to reduce statistical error. At asecond-order transition this is a matter of concern as τA diverges for several observables,especially the order parameter. The divergence of τA is called critical slowing-down.In our simulation we have coped with critical slowing-down by increasing the length ofMarkov chain, ie tN .

So far we have discussed how to estimate statistical error in a single finite-lengthMarkov chain. Another way to further improve data statistics is to carry out mea-surements on several independent Markov chains, respecting the above considerationsfor each of them. In fact it is a way to increase tN in a reasonable manner regardingavailable computing facilities. Note that the effective overall tN is larger than the sumof the individual ones thanks to the independence between different Markov chains.Results presented in this work are typically the average of ten to twenty independentruns.

88 APPENDIX E. MONTE CARLO ALGORITHMS

Appendix F

French summary

Cette thèse de doctorat est consacrée à l’étude d’un modèle paradigmatique de sys-tème magnétique géométriquement frustré, à savoir le modèle antiferromagnétique deHeisenberg sur réseau triangulaire et à ses variantes anisotropes — occasion de proposerun diagramme complet du modèle antiferromagnétique XY sur réseau triangualaire, enrelation avec le cas d’anisotropie de plan facile. Les cas d’anisotropie d’axe facile etde plan facile sont envisagées pour une anisotropie mono-ionique (cf Ann. A) du faitde sa pertinence expérimentale et de son caractère attendu plus altérant que celui del’anisotropie d’échange.

Le chapitre 2 est consacré à la détermination du diagramme de phase du modèleantiferromagnétique de Heisenberg sur réseau triangulaire (Fig. 2.3). Il commence parun rappel sur l’origine du modèle de Heisenberg comme modèle de description de spinslocalisés dans un système cristallin ; suit une discussion des états fondamentaux dumodèle de Heisenberg classique et l’introduction d’une paramétrisation de la structurede spins, Éq. 2.6 (Sec. 2.1). Le comportement à champ nul qui n’a pas fait l’objet d’unenouvelle étude est discuté à travers une revue des résultats obtenus et l’explicationdes lignes d’opposition existant entre les différents travaux et la prise de position enfaveur de l’interprétation proposée par Kawamura d’une transition de phase topologique(Sec. 2.2). La nature des symétries gouvernant le comportement sous champ est ensuiteprésentée, à savoir une symétrie composite S1 ⊗ Z3, avec les implications en terme detransitions de phases (Sec. 2.3). À faible champ, la symétrie discrète Z3 est d’abordbrisée, laissant apparaÓtre un ordre colinéaire, au cours d’une transition qui appartientà la classe d’universalité du modèle vectoriel de Potts à trois états ; la brisure de lasymétrie continue S1 a ensuite lieu lors d’une transition de type BKT vers une phaseà ordre algébrique formée par la configuration à 120 degrés (voir Fig. 2.1). Un élémentmis en avant dans cette étude est qu’a contrario de ce qui se trouvait publié jusqu’alorsles deux lignes de transition à faible champ (brisure de Z3 et de S1) précédemmentdiscutées ne peuvent pas se terminer à un point multicritique à température finie enchamp nul mais doivent rejoindre le point (T, h) = (0, 0) : cela résulte du caractère

89

90 APPENDIX F. FRENCH SUMMARY

sans masse des phases à champ nul par contraste avec le caractère massique de la phasede basse température à champ fini : un champ infinitésimal ne peut pas rendre le systèmemassique. À haut champ la brisure de symétrie est encore successive mais plus difficileà mettre en évidence comme les transitions sont gouvernées par le comportement selonla seule composante transverse pour stabiliser une structure planaire en forme de V(Fig. 2.2). En Sec. 2.4 l’étude numérique réalisée pour construire le diagramme dephase Fig. 2.3 est présentée. Cette présentation commence par une brève esquisse del’algorithme utilisé (cf Ann. E). Sont ensuite introduites les observables utilisées pourlocaliser les transitions, à savoir la chaleur spécifique, le cumulant de Binder, un rapportde fonction de corrélation de spin et la susceptibilité χαq renormalisée selon χαq/L

2−η (cfAnn. D). Enfin les résultats sont explicités.

Dans le chapitre 3 le diagramme de phase du modèle antiferromagnétique de Heisen-berg sur réseau triangulaire avec anisotropie d’axe facile est déterminé (le diagrammeest calculÈ pour d = 0.2, Fig 3.8). Si la symétrie sous champ est identique à celle dumodèle isotrope, celle à champ nul est différente, à savoir S1 ⊗ Z6 : les implicationsen sont discutées en Sec. 3.1. Une présentation en champ moyen est ensuite proposée(Sec. 3.2) pour son apport en termes de discussion des symétries, sa pertinence pourles composés laminaires et l’intérêt du développement d’une théorie correcte de champmoyen dans l’espace réel en présence d’une anisotropie mono-ionique (cf Ann. B & C)qui n’existe pas à ma connaissance dans la littérature et qui se révèle utile par exem-ple pour l’étude de certains systèmes pyrochlores. La détermination du diagramme dephase par une technique Monte Carlo est enfin présentée en deux temps : à champnul (Sec. 3.3) une succession de trois transitions est observée, les deux premières as-sociées à la brisure de la symétrie discrète Z6 avec une phase critique intermédiaire ;sous champ (Sec. 3.4) les brisures de symétries sont similaires au cas isotrope. Dansce dernier cas certaines spécificités apparaissent cependant : la terminaison des lignesde transition au point de transition en champ nul situés à température finie (aux deuxpoints extrémaux en fait), la scission de la transition à température nulle au champde tiers d’aimantation permettant un élargissement de la zone d’existence de la phasecolinéaire autrement réduite à un point dans le cas isotrope ; la réduction du champ desaturation.

Le sujet du chapitre 4 est la détermination du diagramme de phase du modèle anti-ferromagnétique bidimensionnel sur réseau triangulaire : le modèle paradigmatique enest le système de spins XY (Fig. 4.1); dans la continuité de l’étude proposée du modèlede Heisenberg et de ses variantes anisotropes, le cas de l’anisotropie de plan facile estaussi considérée. Après l’introduction des spécificités du modèle XY à champ nul avecl’émergence d’une symétrie chirale, la localisation des deux transitions correspondant àla brisure successive de la symétrie chirale (Z2) puis de la symétrie continue (S1) estprésentée. La nature de la transition chirale est discutée et des arguments sont mis enavant soutenant l’appartenance à la classe d’universalité d’Ising (Sec. 4.1). La déter-mination des transitions sous champ est ensuite explicitée en deux temps. Tout d’abord

91

le cas des champs faibles est étudié (Sec. 4.2) puis celui des champs forts (Sec. 4.3). Àfaible champ l’existence d’un point multicritique auquel les lignes de transition de phases’inverse est mise en évidence : au-dessus de ce point la symétrie discrète compositeZ2 ⊗ Z3 se brise successivement selon la translation du réseau (phase collinéaire) puisla chiralité ; en-dessous de ce point une phase chirale intermédiaire désordonnée mag-nétiquement apparaît qui prolonge celle existant à champ nul ; les lignes de transitionsde phase rejoignent les transitions à champ nul, ce qui est une évidence pour la tran-sition chirale, les transitions étant de même nature. Dans le cas de la ligne associée àla brisure de Z3 cette terminaison est non seulement autorisée comme Z3 est un sous-groupe de S1 mais encore attendue comme le développement de l’énergie libre des ondesde spin à faible champ donne un terme dominant présentant une anisotropie d’ordretrois qui est inessentielle (ou non pertinente) dans le voisinage d’une transition BKT. Àhaut champ l’ordre se fait selon la seule composante transverse ce qui entraîne la brisurede la symétrie comme une symétrie Z6 : ainsi une phase critique intermédiaire existe.Pour compléter l’étude du modèle de Heisenberg et de ses variantes anisotropes, le casdu système avec anisotropie de plan facile qui appartient à la même classe d’universalitéque le système XY est discuté en Sec. 4.4. Il n’y a pas de spécificité particulière, si cen’est une contraction selon l’axe des températures, ce qui est attendu étant donné desfluctuations de spins possibles selon la troisième direction. Le diagramme est présentéen Fig. 4.13 pour le cas d = −0.2.

92 APPENDIX F. FRENCH SUMMARY

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Abstract

This doctoral dissertation presents a thorough determination of the phase diagrams of classical Heisenberg triangular

antiferromagnet (HTAF) and its anisotropic variants based on theoretical and numerical analysis (Monte Carlo). At

finite-field HTAF exhibits a non-trivial interplay of discrete Z3 symmetry and continuous S1 symmetry. They are

successively broken (discrete then continuous) with distinct features at low and high fields: in the latter case the

ordering is along transverse direction; in the former case an intermediate collinear phase is stabilised before 120-degree

structure is. Due to zero-field behaviour, transition lines close at (T, h) = (0, 0).

Single-ion anisotropy is here considered. Easy-axis HTAF for moderate anisotropy strength 0 < d 6 1.5 possesses

Z6⊗S1 symmetry at zero-field which induces triple BKT-like transitions. At finite field the symmetry is the same as for

HTAF: both thus share the same symmetry-breaking pattern. Yet specificities can be observed in the easy-axis system:

splitting of zero-temperature transition at one-third magnetisation plateau, reduction of the saturation field.

Easy-plane HTAF belongs to the class of universality of XY triangular antiferromagnet: it thus interesting to start

with this system. Zero-field behaviour results from the breaking of Z2 ⊗ S1 symmetry, where the discrete component

is an emerging chiral symmetry. An intermediate magnetically chiral ordered phase exists which extends to finite-field

where the symmetry is Z2 ⊗ Z3. The upper limit of this intermediate phase along field axis is a multicritical point at

which transition lines are inverted. Above, the intermediate phase is a collinear phase. At high field the compound

symmetry is broken as a whole Z6.

Résumé

Cette thèse de doctorat présente la détermination théorique et numérique (Monte Carlo) du diagramme de phase

du système classique antiferromagnétique de Heisenberg sur réseau triangulaire (HAFT) et de ses variantes anisotropes.

Sous champ HAFT présente une intrication non triviale des symétries discrète Z3 et continue S1. Elles sont succes-

sivement brisées (discrète puis continue) selon des modalités différentes à champ fort et modéré : dans ce cas-là l’ordre

a lieu selon la direction transverse ; dans ce cas-ci une phase colinéaire intermédiaire est stabilisée avant la phase à

120 degrés. Du fait du comportement à champ nul les lignes de transitions se terminent à (T, h) = (0, 0).

L’anisotropie mono-ionique est ici considérée. HAFT avec anisotropie d’axe facile pour une anisotropie modérée,

0 < d 6 1.5, possède une symétrie Z6⊗S1 à champ nul, qui induit une triple transition BKT. Sous champ, la symétrie

est identique à HAFT : les deux partagent donc le même scénario de brisure de symétries. Le système anisotrope

présente toutefois des spécificités ; séparation de la transition à température nulle au champ de tiers d’aimantation,

réduction du champ de saturation.

HAFT avec anisotropie de plan facile appartient à la classe d’universalité de XY AFT il est donc intéressant de

commencer par ce système-ci. Le comportement à champ nul résulte de la symétrie Z2 ⊗ S1 où la composante discrète

est une symétrie chirale émergente. Une phase intermédiaire chirale magnétiquement désordonnée est stabilisée ; elle

se prolonge sous champ, où la symétrie est réduite à Z2 ⊗ Z3, jusqu’à un point multicritique auquel les transitions

s’inversent. Au-dessus de celui-ci la phase intermédiaire est colinéaire. Sous champ fort la symétrie composite se brise

comme une symétrie Z6 unique.

Keywords: statistical physics, frustrated magnetism, triangular antiferromagnet, Monte Carlo, over-relaxation